Measuring Fluorescence to Track a Quantum Emitter’s State: A Theory Review

Philippe Lewalle,1, 2, ∗ Sreenath K. Manikandan,1, 2 Cyril Elouard,1, 2 and Andrew N. Jordan1, 2, 3 1Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627, USA 2Center for Coherence and Quantum Optics, University of Rochester, Rochester, NY 14627, USA 3Institute for Quantum Studies, Chapman University, Orange, CA 92866, USA (Dated: May 29, 2020) We review the continuous monitoring of a qubit through its spontaneous emission, at an introduc- tory level. Contemporary experiments have been able to collect the fluorescence of an artificial atom in a cavity and transmission line, and then make measurements of that emission to obtain diffusive quantum trajectories in the qubit’s state. We give a straightforward theoretical overview of such scenarios, using a framework based on Kraus operators derived from a Bayesian update concept; we apply this flexible framework across common types of measurements including photodetection, homodyne, and heterodyne monitoring, and illustrate its equivalence to the stochastic master equa- tion formalism throughout. Special emphasis is given to homodyne (phase–sensitive) monitoring of fluorescence. The examples we develop are used to illustrate basic methods in quantum trajectories, but also to introduce some more advanced topics of contemporary interest, including the arrow of time in quantum measurement, and trajectories following optimal measurement records derived from a variational principle. The derivations we perform lead directly from the development of a simple model to an understanding of recent experimental results.

I. INTRODUCTION [12, 56]. Thus, in contrast with the closed–system quan- tum mechanics described by the Schr¨odingerequation The literature on quantum theory and quantum optics alone, a physical description of the measurement process is replete with works concerning the spontaneous emis- necessarily requires that we consider our primary system sion of atoms, across virtually all of its century–long his- as being open. A “quantum trajectory” arises when a se- tory [1–8]. The generic case, in which the excited–state quence of measurements are made in time, such that we population of an emitter decays exponentially on aver- have a time–series of measurement outcomes, and a cor- age due to the spontaneous emission of a photon, is a responding time–series of inferred quantum states of our paradigmatic phenomenon in quantum optics. More re- primary system, based on that information. The process cently, both the theory [9–41] and experiments [42–46] is necessarily stochastic, as there is randomness present about continuous quantum measurement have received in each successive measurement outcome. considerable attention, and seen rapid progress, reveal- Our emphasis will be on tracking the state of a qubit ing new phenomena and insights into the quantum mea- (a two–level quantum system), through its spontaneous surement process [46–51], and applications to quantum emission; this means that the qubit is coupled to a field control [52–54]. In any such generalized measurement(s) mode, which is its “environment” in this scheme, and [12, 55] of some primary system of interest, there must by interrogating this mode in a variety of ways, we will necessarily be some series of interactions between that be able to infer a corresponding evolution of the qubit’s system and its environment, which allows for informa- state. A quantized electromagnetic field mode is repre- tion to flow from the primary system to some meter(s) sented by a quantum harmonic oscillator; we will dis- which record the measurement outcome(s) [56]. The in- cuss the cases where we interrogate the output field by teraction between the system and environment will nec- photodetection (effectively an energy measurement), or 1 essarily disturb the system of interest in a random way, by quadrature measurements (homodyne or heterodyne but inferences about that evolution of the system of in- detection are analoguous to making “position” and/or terest can be drawn as long as our measurement brings us “momentum” measurements of the oscillator). We are arXiv:1908.04720v3 [quant-ph] 28 May 2020 the information that the environment “learned” by inter- motivated in large part by recent experimental work, in acting with the system [57]. Generalized measurements which a superconducting transmon qubit is continuously can be weak (a small amount of information is acquired monitored by homodyne or heterodyne detection of its about the system state, with correspondingly little dis- spontaneous emission, leading to diffusive quantum tra- ruption to its prior behavior) [58–62], or strong (e.g. the system is “collapsed” to an eigenstate of the measure- ment operator by a projective measurement, such that we 1 The quadrature space of the field is effectively the phase space have acquired a lot of information at once and disturbed of the quantum harmonic oscillator describing the field mode in the state by corresponding ramifications in the process) question. In other words, a quadrature is analoguous to the “po- sition” or the “momentum” of a quantum harmonic oscillator, and the product of the noise in orthogonal directions in quadra- ture space is bounded by the Heisenberg uncertainty principle. A reader unfamiliar with a quadrature phase space representation ∗ [email protected] of a field mode may benefit from perusing e.g. Refs. [63, 64]. 2 jectories [24, 33, 65–73]. Following the relevant circuit– (a) (b) QED experiments, the physical setup we have in mind throughout this work involves a single qubit placed in- side a cavity, such that microwave photons emitted by the qubit via spontaneous emission are coupled into a transmission line leading to a measurement device. The hom.∼phase–sensitive (c) setup is designed such that photons emitted by the qubit via spontaneous emission are transmitted to the detec- Ω tor, but photons in other modes (e.g. to implement some QLA r unitary Rabi rotations on the qubit) are not routed to- LO,θ wards it. Such devices allow for high collection efficiency het.∼phase–preserving of emitted photons, in contrast with situations in which an atom emits into free space. See Fig.1 for an illustra- LO,θ rI tion, and guides to the experimental details can be found balanced QLA LO,θ e.g. in Refs. [67–75]. dyne rQ We will proceed by splitting our manuscript into two main parts. In the first part, we describe a qubit open to √ (d) ˆ† a decay channel and subsequent measurements from sev- 1 − η a` eral different perspectives. We carry out the formal treat- aˆ† any ideal ment of an unmonitored decay channel, using first the √ † detector typical quantum–mechanical analysis in Sec.IIA, and η aˆs then by introducing the corresponding master equation in Sec.IIB. We transition towards diffusive trajectories in vac. Sec.III, introducing the Kraus operators that will serve as our primary tool, along with the Stochastic Master FIG. 1. We show schematics representing the possible ex- Equation (SME). We examine the cases of heterodyne or perimental situations we consider. Historically, treatments homodyne detection in detail, in sectionsIV andV, re- of spontaneous emission have often focused on the average spectively. We first discuss ideal measurements in which dynamics of emitters, without any monitoring of individual all information is collected, and then describe inefficient emitters and their individual emission events, represented in measurements in which some information is collected and (a). Our emphasis here is on devices in which the emission is some is lost in sec.VB. Each of these steps is repre- captured in a cavity / transmission line, and routed to a de- sented graphically in Fig.1. In the second part, we use tector, such as (b) a photodetector, or (c) a homodyne or het- erodyne setup which measures one or both of the signal field’s the examples developed in the first part as a springboard quadratures. Optically, the single–photon signal is mixed to introduce certain concepts and methods of interest in with a strong local oscillator (LO) on a 50/50 beamsplitter for the current research literature. For example, we are able quadrature detection, leading to a readout of one (homodyne) to use these examples to introduce ideas related to the or both (heterodyne) quadratures of the field. Contemporary arrow of time in quantum trajectories in Sec.VIA, and circuit–QED experiments using microwave photons typically discuss “optimal paths” (OPs) [32, 34, 47, 50, 66, 76] perform these measurements using a quantum–limited ampli- in Secs.VIB andVIC, which are quantum trajectories fier (QLA) built from Joesphson junctions (see e.g. [28, 77– which connect given states according to an extremal– 79]). The measurement axis in the quadrature phase space is probability readout, derived according to a variational determined by the relative phase θ between the signal and LO principle. Summary, outlook, and further discussion are / amplifier pump tone. Relevant experiments often include a included in Sec.VII. drive characterized by the Rabi frequency Ω. In (d) we illus- trate a simple model for measurement inefficiency, in which an unbalanced beamsplitter splits the ideal signal into a mea- sured portion with probability η ∈ [0, 1] and a lost portion II. UN–MONITORED DECAY (wherea ˆ† denotes a photon creation operator). See Sec.VB for further details on this last point. We review the case of a single qubit whose fluorescence goes unmonitored in two parts. First we review the stan- dard treatment of Weisskopf and Wigner [3]; next we “artificial atoms” now used in many experiments. The introduce an equivalent master equation description of state of a qubit can be represented as living in the Bloch the system [55, 57, 80]. sphere; we will generically parameterize our single–qubit A qubit is any two–level quantum system; density matrix with Bloch coordinates according to mathematically–speaking this means that it is de-   scribed like a spin– 1 . Physically speaking, a qubit might 1 1 + z x − iy 2 ρ = = 1 (11 + xσˆ + yσˆ + zσˆ ) be any of e.g. a particular transition in an atom or ion, 2 x + iy 1 − z 2 x y z a spin in a quantum dot or diamond nitrogen–vacancy (1) center, or the lowest–two levels of the superconducting throughout the forthcoming derivations, where (1 + 3

z |ei 1 (a) (b) N z y

• |ψi hψ|

ϑ

℘Υ x −1J I1

x • ρm = ℘ψ |ψi hψ| +℘Υ |Υi hΥ|

℘ψ |gi • |Υi hΥ| H −1

FIG. 2. We show the Bloch sphere (a), with |ei at the “north pole” (z = 1), and |gi at the “south pole” (z = −1). Any pure qubit state can be represented by a point on the sphere’s surface, while mixed states (i.e. weighted averages of two or more pure–state density operators) live inside the sphere’s surface. In (b), we highlight some of these features in more detail, focusing on the xz–plane of the Bloch sphere. Some special states |ei = , |gi = , √1 (|ei − |gi) = , and √1 (|ei + |gi) = N H 2 J 2 I are marked. A mixed state ρm is shown as the weighted sum of two pure state density operators ρψ and ρΥ(where the vectors drawn are proportional in length to the probability of drawing their respective pure state from an ensemble). We also illustrate the coordinate ϑ used later in the text.

z)/2 = ρee denotes the excited state population. See By using the density matrix to describe our state, we may Fig.2. It is also necessary that we introduce a dis- account for all of these options, which is necessary when tinction between pure states, which can be represented we have an open system and the possibility of information by a state vector |ψi, and mixed states which require loss. the use of ρ. Pure qubit states ρ = |ψi hψ| live on the Note that ρ is Hermitian (ρ = ρ†), and normalization outer surface of the unit sphere, while more general states requires that tr (ρ) = 1. We will often represent a qubit’s P ρ = i ℘i |ψii hψi| may live inside the sphere; a “mixed” state and dynamics in terms of the Bloch vector q = state with more than one non–zero ℘i may be used to {x, y, z} below; such coordinates should be understood describe a system which is imperfectly isolated from its in the context of (1). We suppress the “hat” notation on surrounding environment, where the ℘i are effectively our density operators ρ. probabilities. This description implies that we have a ’classical’ statistical mixture, in which we have a prob- ability ℘i of finding the pure state |ψii in an ensemble; A. Standard quantum–mechanical treatment in contrast with a coherent pure state superposition, ele- ments in such a mixture do not interfere with each other. We summarize the typical approach, originally by For example, we may consider states which lead to a Weisskopf and Wigner, as a point of departure in de- 1 probability 2 for a σz–measurement to return |ei or |gi, scribing spontaneous emission. A more complete deriva- such as |x+i = √1 (|ei + |gi). The density matrix for this tion of the results we summarize can be found in Ref. [27], 2 pure state (which contains the possibility for quantum and we will stick closely to their conventions for clarity. interference) is not the same as the classical statistical The Hamiltonian describing the joint system including mixture of |ei and |gi (where the off–diagonal “coher- the qubit and a single mode of the electromagnetic field ences” are suppressed), i.e. is of the form hˆ = ω σˆ+σˆ + ω aˆ†aˆ + 1
+ gσˆ+aˆ + g∗σˆ aˆ†
,   qb − f 2 − (3) 1 1 1 ~ | {z } | {z } | {z } |x+i hx+| = qubit single field mode ˆ 2 1 1 interaction hint   (2) 1 1 0 1 1 where ~ωqb is the energy separation between the two 6= = 2 |ei he| + 2 |gi hg| . 2 0 1 qubit levels of interest, ωf denotes the frequency of the 4

field mode, and g is a coupling constant between the field and qubit. The first two terms represent the qubit and field, respectively, while the third term describes their interaction. One can regard this model as corresponding to a two level system and quantum harmonic oscillator which are able to exchange excitations. We have rais- + ing and lowering operators on the qubitσ ˆ =√|ei hg| and σˆ− = |gi he|,√ and on a field modea ˆ |ni = n |n − 1i anda ˆ† |ni = n + 1 |n + 1i for Fock states |ni. The interaction term ~gσˆ+aˆ describes the possibility for the atom to become excited by absorbing a photon (which is removed from the field); the adjoint of this term de- notes the reverse process, in which the qubit loses en- ergy, emitting a photon which is added to the field mode. We may use the Schr¨odingerequation to compute the evolution of the joint system and subsequent qubit state amplitudes under the influence of this Hamiltonian. The electro–magnetic environment of the qubit contains many modes, and the apparently incoherent evolution of the qubit associated with spontaneous emission emerges when summing up the action of the couplings to all these ˆ modes, each described like hint above. Such analysis re- quires that we average the evolution predicted by the t = 0 t = 5T1 Schr¨odingerequation over the available density of free– space field modes and sum over polarizations [27]. To FIG. 3. We plot the evolution of the qubit state under the good approximation (i.e. the approximations first made unmonitored fluorescence dynamics (10) in the xz–plane of by Weisskopf and Wigner), we may simplify the dynamics the Bloch sphere, originating from a variety of initial pure at timescales much longer than the field periods, elimi- states. The excited state is at the top of the sphere, and nating environmental modes from the description. For a all paths converge towards the ground state at the bottom. generic qubit state ζ |ei + φ |gi, we obtain the evolution Color denotes the time evolution along each path. The tra- jectories in the qubit state tend to become impure / mixed, of the excited state amplitude because no information is collected about fluorescence out- ˙ γ
ζ = −iωqb − ζ. (4) put; we have an open system with lost information in this 2 case. These dynamics are often represented with an equiva- We have introduced the spontaneous emission rate γ = lent picture in which the Bloch ball contracts into an ellipsoid −1 T1 which is equal to the density of modes coupled reso- near the ground state under the influence of a decay channel P 2 [55]; we sample a greater number of initial states at intervals nantly to the qubit via γ = j |gj| δ(ωqb −ωj). Here, gj stands for the strength of the coupling to the jth mode of T1/2 to illustrate this. of the electromagnetic reservoir. The contributions of all the non–resonant modes oscillate and quickly average to −1 B. Master equation treatment zero over a typical time τcorr  γ ; this condition is key in allowing us to obtain a Markovian description of the qubit evolution (4), from which the reservoir dynamics In order to express the complete information about the have been completely eliminated. A precise discussion of qubit state at any time in a compact way, and straight- the approximations leading to Eq. (4) and the order of forwardly generalize our system (e.g. we might consider the associated errors can be found in Ref. [81], Chapter coherently driving qubit), it is convenient to formulate 4. the dynamics of spontaneous emission as a master equa- The excited state population is described by the den- tion for the density operator ρ introduced above. The sity matrix element evolution of an open quantum system in contact with a Markovian environment (i.e. with an environment of very ∗ ρee = ζ ζ = (z + 1)/2. (5) short correlation time τcorr with respect to the other time It is straightforward to compute that (4) and (5) imply scales in the problem) can, in general, be written as a Lindblad equation; such a master equation is of the form ρ˙ee = −γρee ↔ z˙ = −γ (z + 1) . (6) [57] The result that spontaneous emission leads to exponen-   i ˆ X ˆ ˆ† 1 ˆ† ˆ 1 ˆ† ˆ tial decay of the excited state population at rate γ (or ρ˙ = [ρ, H] + LcρLc − 2 LcLcρ − 2 ρLcLc , −1 ~ c with characteristic time T1 = γ ), absent other dynam- | {z } Unitary Evol. | {z } ics, is among the most fundamental phenomena in the Lindblad Dissipation quantum optics literature. (7) 5 where ρ is the density matrix of the system of primary damping is drastically modified. In Ref. [83] this effect interest (in this case, the qubit), and each operator Lˆc was exploited to stabilize an arbitrary state of the Bloch describes a coupling between the system and its environ- sphere. However, provided the drives are weak enough ment (in this case, the decay channel). We see a term (Rabi frequencies much smaller than the qubit frequency i describing the unitary evolutionρ ˙ = [ρ, Hˆ ], plus the and the inverse correlation time of the reservoir τcorr), Lindblad term which accounts for information~ leaking and there is no cavity or other resonance close to the into the environment through any channels to which the driven qubit’s emission spectrum peaks to cause espe- system is open. cially fast variations in the environment spectrum [84], The arguments of the previous section can be used to this effect is negligible. Within these conditions, the ac- show that the case of spontaneous emission corresponds tion of a drive can be simply captured by adding a uni- √ ˆ to a single channel characterized by operator Lˆ = γσˆ− tary term (i/~)[ρ, Hdr(t)] in Eq. (8); the treatment we [81], which indicates that the qubit may lose its excitation develop below assumes this simplest case. While some with an effective coupling rate γ. The master equation modification to this simplest scheme may be necessary in capturing spontaneous emission of a qubit is then adapting it to situations beyond the stated constraints, experiments not explicitly focused on engineering more i + γ + +
exotic effects will typically obey these simplifying con- ρ˙ = [ρ, Hˆ ] + γσˆ−ρσˆ − σˆ σˆ−ρ + ρσˆ σˆ− , (8) ~ 2 straints by default; this simplest scheme we lay out below is thus widely applicable. ˆ where the qubit Hamiltonian is H = ~ωqbσˆz/2. The unitary part solely induces a rotation at frequency ωqb of the qubit state around the z-axis of the Bloch sphere, III. QUANTUM TRAJECTORIES and it is convenient to work in a frame in which this rotation is suppressed. This rotating frame is formally The treatment of spontaneous emission in the previous the interaction picture with respect to Hˆ , associated with section, and in particular the master equation Eq. (9), iHt/ˆ −iHt/ˆ the transformation ρ → e ~ρe ~. In the following, captures the dynamics of the qubit under the assumption we will always work in such frame, where the master that any information emitted by the qubit (leaking into equation reads the environment) during the qubit-field mode interaction is lost forever. We are, however, primarily interested in ρ˙ = γσˆ ρσˆ+ − γ σˆ+σˆ ρ + ρσˆ+σˆ
. (9) − 2 − − the case where we, the observer(s), recover some (or ide- We can get equations of motion in the Bloch coordinates ally all) of this information through measurement(s) on the field mode. In this section, we present the formalism (in the rotating frame) by computing q˙ = tr(ˆσqρ˙), yield- ing of Kraus operators, which describes the update of the qubit’s state conditioned on acquiring such information. γ γ x˙ = − x, y˙ = − y, z˙ = −γ(1 + z), (10) 2 2 A. Kraus Operator Formalism in perfect agreement with the treatment above (6). The decaying solutions of these equations, initialized from dif- ferent pure states on the edge of the Bloch sphere, are The basic idea is that there exist a set of Kraus op- ˆ illustrated in Fig.3. erators Mr, which describe how the state of our system For many application, the qubit needs to be driven; should be updated, each of them conditioned on acquir- to describe such a situation, we must modify the qubit ing one of the possible measurement outcomes r in the Hamiltonian Hˆ , adding time–dependent terms. In gen- environment during a measurement of duration dt, ac- eral, the derivation of the master equation describing the cording to [12, 55, 57] dynamics of the qubit’s density operator needs to be care- Mˆ ρ(t)Mˆ † fully redone in presence of this new Hamiltonian, and ρ(t + dt) = r r . (11)  † one may find that the Lindblad term is modified due to tr Mˆ rρ(t)Mˆ r the presence of the drive [82]. Such a modification typi- cally occurs, for example, when the drive causes the qubit We require that either P Mˆ †Mˆ = 11, or R drMˆ †Mˆ = to become sensitive to modes of the environment at dif- r r r r r 11, depending on whether the possible measurement out- ferent frequencies, which are sufficiently separated from comes r are discrete or continuous; such a condition tells each other so as to have a different density of states. For us that we have a valid (completely positive) transfor- instance, in the case of a quasi–resonant monochromatic mation on ρ, and insures that we have considered a drive inducing Rabi oscillations, the emission spectrum complete, self–consistent set of measurement outcomes. of the qubit contains multiple peaks (the famous Mol- As dictated by the axioms of quantum mechanics, the low triplet [4]) separated by the Rabi frequency (which outcome r is obtained randomly among its possible val- is related to the intensity of the driving). If the envi- ues based on Born’s rule, which here yields probabilities ronmental density of state varies around the qubit fre-   ˆ ˆ † quency ωqb on the Rabi frequency scale, the form of the (or a probability density) ℘(r|ρ) = tr MrρMr . Note 6 that the denominator of (11), which serves to ensure the where {|ψri} could be any basis of states of the field updated density matrix is properly normalized, exactly mode, which should be chosen based on the kind of mea- matches this probability. If measurement(s) on the en- surement being performed and result r. All of the exam- vironment is(are) repeated (in our case every dt), the ples we consider below rely on a Kraus operator of the successive outcomes and subsequent state updates define form (15). Much of what we do below will revolve around a stochastic sequence of states called a quantum trajec- relating different measurements to the appropriate choice tory. of |ψri, and then exploring the ramifications that choice Following Ref. [66], we construct the particular Mˆ of has on the measurement backaction and quantum trajec- interest for the case of a spontaneously–emitting qubit, tories. using a Bayesian probability argument; it is useful to consider a pure state of the qubit and an effective field mode it emits into (initially in vacuum state |0i) B. Photodetection and quantum jump trajectories

|ψ0i = (ζ |ei + φ |gi) ⊗ |0i , (12) As a first example, suppose that we choose our |ψri in where z = 2|ζ|2 − 1, and |ei and |gi are the excited the Fock basis, i.e. we consider outcomes of the type |1i and ground states of the qubit, respectively. There is (a photon exits in the field mode in the given timestep), a probability ℘(e) = |ζ|2 to find the qubit in |ei and or |0i (no photon exits), which correspond to making a probability ℘(g) = |φ|2 to find the qubit in |gi, with photodetection measurement. In other words, we imag- ℘(e) + ℘(g) = 1. On phenomenological grounds, we ine counting the photons emitted by the qubit into the field mode, in a time–resolved manner, with a detector suppose that the probability for an emission event in a −1 time interval dt is given by ℘(1|e) =  = γ dt, where γ integration time dt  γ (equivalently,   1). ˆ ˆ is some characteristic rate at which the qubit fluoresces We may define Kraus operators M1 (M0) for the single–qubit state update conditioned on a click (no– (i.e. γ = 1/T1 is a measurable quantity for a qubit–cavity system). Then ℘(1|e)℘(e) = |ζ|2 = ℘(e|1)℘(1) and/or click) in the detector, according to 2 ℘(0|e)℘(e) = (1−)|ζ| = ℘(e|0)℘(0) according to Bayes’  √   √  theorem. A quantum–coherent state assignment after the ˆ 1 −  0 1 −  0 M0 = h0| √ † |0i = , (16) short interval dt which reflects these probabilistic consid- aˆ 1 0 1 erations is √ √ √  1 −  0   0 0  |ψ1i = 1 − ζ |e, 0i + φ |g, 0i + ζ |g, 1i . (13) Mˆ = h1| √ |0i = √ . (17) 1 aˆ† 1  0 In other words, there is some probability for an emission event which involves a photon being created in the out- ˆ † ˆ ˆ † ˆ It is easy to verify that M0 M0 + M1 M1 = 11, such that put mode (0 → 1), and which shifts qubit population these measurement operators form a positive operator from |ei → |gi, reflecting a common sense understanding valued measure (POVM) [55]. We can say that under of spontaneous emission. Below we will always assume continuous photodetection, the qubit state is updated ev- that the measurement time (in practice, a detector inte- ery dt by gration time) is much faster than the characteristic decay † time of the qubit, i.e. we have dt  T1, or   1. This Mˆ ρ(t)Mˆ ρ(t + dt) = 1 1 , (18) is a key condition which will ensure that the quadrature  ˆ ˆ † measurements we will eventually consider are weak mea- tr M1ρ(t)M1 surements, and that the subsequent quantum trajectories are diffusive. We also assume that the information we ac- if the detector registers that a photon emerged between quire applies to the qubit in real time, which implies that t and t + dt, or according to that photon travel time between the qubit and measure- Mˆ ρ(t)Mˆ † ment apparatus should be negligible. We may rewrite ρ(t + dt) = 0 0 (19) the change of state from above as  ˆ ˆ † tr M0ρ(t)M0 √  1 −  0   ζ  |ψ i = √ |ψ i for |ψ i = ⊗|0i , (14) if no photon reaches the detector. The probability of 1 † 0 0 φ aˆ 1 ˆ ˆ † a click in any given timestep is ℘1 = tr(M1ρ(t)M1 ) = where a† creates a photon in the relevant cavity/field γ dt (1 + z)/2, and the probability of no–click is ℘0 = † † tr(Mˆ ρ(t)Mˆ ), with ℘ + ℘ = 1. These expressions re- output (a |0i = |1i). The Kraus operators Mˆ r in (11) act 0 0 0 1 only on the qubit state, and are obtained by projecting flect the common–sense result that ℘1 must vanish when out the field mode in a final state corresponding to some the qubit is in the ground state, i.e. ℘1 = 0 for z = −1. outcome from measuring the field, i.e. Thus, a single quantum trajectory for this photodetec- √ tion scenario is characterized by a time series of out-  1 −  0  comes r ∈ {0, 1}. Simulation of such a trajectory can Mˆ = hψ | √ |0i , (15) r r aˆ† 1 be performed by drawing a click/no–click readout from a 7 binomial distribution at each short timestep of duration generically be obtained by expanding an expression of dt  T1, and subsequently updating the qubit state ρ ac- the form (11) to O(dt)[25, 27] (detailed examples of this cording to the appropriate rule above. Results of such a process follow below). The addition of a stochastic ele- simulation are shown in Fig.5(a). The trajectories gen- ment into a differential equation is not trivial, because a erated by photodetection are an example of “quantum genuinely stochastic element is not really differentiable, jump” trajectories, for which the qubit state immediately the way a smooth and well–behaved function is. jumps to |gi when a click event occurs (this is related to Generically, what we will momentarily consider is a the discrete nature of the possible outcomes r). type of Langevin equation, or first–order stochastic dif- Before moving on to different types of measurements ferential equation of the form on the output mode, we bridge the gap between our Kraus operator description and the un-monitored decay q˙ = a(q) + b(q)ξ(t); (22) channel we discussed in the previous section. The situa- the term a is often called the drift term, whereas b tion in which the outcome of the measurement performed functions as a diffusion constant, and together with the on the field mode between t and t+dt is actually unavail- randomly–varying ξ(t), gives stochastic evolution. Equa- able can be captured by averaging the state update over tions of this type were first written down to model Brow- both outcomes, i.e. nian motion of small particles [85], where complex me- chanical forces lead to effectively random kicks in a par- Mˆ ρ(t)Mˆ † + Mˆ ρ(t)Mˆ † ρ(t + dt) = 0 0 1 1 . (20) ticle’s position. In our present case, we care about the  ˆ ˆ † ˆ ˆ † tr M0ρ(t)M0 + M1ρ(t)M1 evolution of a quantum state, and the stochasticity de- noted by ξ(t) is a result of the randomness inherent An equation of motion can be obtained by taking in the quantum measurement process. The particular ρ(t + dt) − ρ(t) type of random evolution we consider is delta–correlated ρ˙ ≈ , (21) dt Gaussian white noise, obeying ξ(t) = dW (t)/dt, where W (t) is called a Wiener process. The Wiener increment where the numerator on the RHS is expanded to O(dt). dW (t) = W (t + dt) − W (t) is a Gaussian random vari- It is then straightforward to verify that the equations (10) able, independent on any past values dW (s) for s < t reappear exactly, i.e. the procedure just described to ob- and characterized by a mean of zero and variance equal tainρ ˙ leads to exactly the same expression as the master to dt. These properties lead to a noise term ξ(t) of zero equation as described above, and as shown in Fig.3.A expectation value hhξ(t)ii = 0 and co-variance obeying similar procedure allows to show that for any measure- hhξ(t)ξ(t0)ii = δ(t − t0), where the double bracket indi- ment basis |ψri chosen for the field, the master equation cates the ensemble average over realizations of the pro- is recovered when averaging over all of the outcomes we cess. This is suitable for describing the quantum noise could have obtained from measurement; we will soon be arising from measurement in a variety of physical situa- able to elaborate further on this point. tions, including those we consider below3. Some physical justification for the appropriateness of the use of a Gaus- sian ξ for the examples below is provided in the following C. Diffusive trajectories and stochastic master equation sections, and in AppendixA. For a = 0 and constant b (simple diffusion without drift), the variance of an ensem- ble of diffusing trajectories scales like time; this is sum- In the remainder of this article, we are concerned marized by the Itˆostochastic calculus rule dW (t)2 = dt about measurements on the environment leading to a (or equivalently ξ(t)2 = 1/dt). continuous–valued outcome r, e.g. a voltage or current The general form of the SME that we use for diffusive from a detector, leading to “diffusive” trajectories (in quantum trajectories, in units → 1, reads [12, 27] contrast with the “jump” trajectories we have just dis- ~ cussed). The specifics of the two most common examples, X  √  dρ = i[ρ, Hˆ ]dt + Lˆ[ρ, Lˆc]dt + ηcMˆ [ρ, Lˆc]dWc . heterodyne and homodyne measurements, are presented c in detail in the following section. Because the evolu- (23) tion during dt is infinitesimal, it is common to write the The super–operators are the Lindblad dissipation term, change in the density operator of the qubit, conditionned from (7), on the outcome r obtained at time t, under the form of a   ˆ ˆ ˆ ˆ† 1 ˆ† ˆ ˆ† ˆ stochastic master equation (SME); the SME can be seen L[ρ, Lc] ≡ LcρLc − 2 LcLcρ + ρLcLc , (24) as an extension of Eq. (7), in which we add a term which accounts for the measurement outcome2. The SME may

3 Strictly speaking, writing ξ(t) = dW (t)/dt is an odd mathemati- cal statement, because W (t) is pure noise and non–differentiable. 2 Photodetection, as considered above, constitutes a particular In practice such substitutions does not cause us a problem in “unraveling” of the master equation into stochastic trajectories; writing down sensible stochastic calculus however. For details, the heterodyne and homodyne measurements we subsequently refer e.g. to the books by Gardiner [10, 86], or other references consider are additional possible “unravelings”. on stochastic differential equations, such as [87]. 8 and the newly–added measurement backaction term q q

ˆ ˆ ˆ ˆ† ˆ ˆ† M[ρ, Lc] ≡ Lcρ + ρLc − ρ tr Lcρ + ρLc . (25) t t As before, a Hamiltonian Hˆ may describe any unitary q q processes applied to the system (e.g. Rabi drive on a qubit). Each of the operators Lˆc describes a particu- lar measurement channel, which is monitored with effi- t t ciency ηc ∈ [0, 1] (where 1 denotes perfect measurement efficiency, and ηc is dimensionless). The measurement record associated with any monitored channel, contains FIG. 4. We illustrate some of the concepts implicit in (26). √ † one outcome every dt, going like rc ∝ ηchLˆc + Lˆ i + ξc, The Itˆo–like choice β = 0 is illustrated with red boxes in c 1 where the brackets denote the expectation value in state all subfigures, while the Stratonovich–like choice β = 2 is ρ. Such an expression for the readout is easy to interpret illustrated with the blue trapezoids in all subfigures. We see as a signal hLˆ + Lˆ†i, attenuated due to inefficiency by a these applied to an ordinary differential equation (b = 0) on √ c c the left, and to a stochastic differential equation (b 6= 0) on factor ηc, plus quantum noise ξc intrinsic to the mea- the right, with timesteps decreasing as we go from the top surement process. A more detailed introductory guide to down. All choices of β will converge to the same area under SME can be found in Ref. [27]. the curve in the time–continuum limit for smooth function Channels which are open to the environment, but un- on the left. For the stochastic process depicted on the right, monitored (e.g. typical dephasing mechanisms, or the de- however, (which remains stochastic at any timescale, such cay channel in the un-monitored case), can be modeled by that the kind of picture on the left never emerges from it), placing an operator in the sum over c which is monitored different choices of β will not necessarily lead to the same solution. Some ramifications of this are discussed in the main with efficiency ηc = 0. The master equation (7) can be recovered from the SME by taking an ensemble average text, leading to equations (27) through (29). over stochastic trajectories Applying these concepts to the example of a single decay channel introduced above, where we haveq ˜k = βqk+1 + (1 − β)qk, and the indices k, we see that 1) opening the qubit to an unmonitored de- k + 1 correspond to times ∆t apart4. We highlight two ˆ √ cay channel L = γσˆ−, 2) measuring the qubit fluores- very common conventions: The Itˆoconvention uses β = √ cence/decay according to Lˆ = γσˆ− with efficiency zero, 0, such that we evaluate drift and diffusion coefficients or 3) the average dynamics over an ensemble of stochas- at the beginning of a timestep, whereas the Stratonovich tic trajectories obtained by continuously monitoring the convention corresponds to β = 1 , such that functions are √ 2 qubit fluorescence as per Lˆ = γσˆ−, are all equivalent evaluated according to a trapezoidal rule (see Fig.4). situations. This view from the master equation is also The form of the SME (23) assumes a derivation based entirely equivalent to that for the Kraus operators, as on Itˆocalculus, in which expansions are made to O(dt) presented in and around (20). In Fig.5 we observe this in using the rule dW 2 = dt (i.e. expansions to O(dt) must simulations, observing that the average over many quan- include explicit expansions to O(dW 2) in that formalism tum trajectories reproduces the dynamics of the unmon- [27]). For an accessible and intuitive explanation of this itored case (10), regardless of the character of the indi- rule, we encourage the interested reader to look at section vidual measurements. 4 of Ref. [37]. Expansions made with regular calculus In the sections below, we will formallly compare equa- will lead a Stratonovich equation instead however. In tions of motion derived from our Kraus operator meth- other words, we have to consider two different stochastic ods to those from the SME; in order to do this, it is calculus conventions, each leading to different differential necessary that we briefly comment on a technical issue equations; they give consistent results, however, when pertaining to stochastic calculus and the integration of paired with the correct integration rules. Specifically: stochastic differential equations. The calculus used to Integrating the equation derive and/or manipulate a Langevin of the type above dq = a(q) dt + b(q) dW (27) is closely tied to the type of Riemann sum used as the basis of any subsequent integration. If we were integrat- according to the Itˆosense (β = 0), is equivalent to per- 1 ing an ordinary differential equation, any valid choice of forming a Stratonovich integration (β = 2 ) on Riemann sum would lead to the same result in the time– q˙ = A(q) + b(q) ξ, (28) continuum limit. This is not so in the stochastic case however; if we suppose that dW is stochastic at every time–scale, different Riemann sums will not converge to 4 R t 0 0 0 R t 0 0 0 Formally, q(t) − q(0) = 0 dt a[q(t ), t ] + 0 dW (t )b[q(t ), t ] is the same solutions in the limit anymore! To get the idea, more appropriate; the way the integration of the diffusion term, we may consider a discrete update step over dW , is carried out is both significant and potentially am- biguous. See chapter 4 of [86] for rigorous derivations and more detailed comments. qk+1 − qk = a(˜qk)∆t + b(˜qk)∆Wk, (26) 9 where the two drift terms a and A are related by the A. Stochastic Master Equation Treatment transformation The SME is given in Eq. (23), and provides one of 1 X Aq = aq − 2 bjn∂nbjq; (29) the most–used approaches to modeling diffusive quan- j,n tum trajectories arising from continuous weak measure- ment [25, 27]. We will consider an idealized measurement n indexes the coordinates (components of q), and j in- in the rotating frame, characterized by Hˆ = 0 (no uni- dexes the independent noise(s) on each measurement tary dynamics), Lˆ =σ ˆ pγ/2, and Lˆ = iσˆ pγ/2, channel, which are summed. For justification and details X − P − where there is no dephasing channel and the measure- see e.g. [86, 87]. We will use this conversion rule to con- ment efficiency η = 1 is perfect. We can make qualitative nect different descriptions of the quantum measurement sense of the two operators Lˆ and Lˆ by understanding scenarios we consider below. X P thatσ ˆ indicates that our measurement is being made We make a final remark about numerical simulations − through a decay channel, and that Lˆ and Lˆ are asso- before moving on. The appeal of the SME as a theoret- X P ciated, respectively, with the information encoded in the ical tool is that it expresses quantum trajectory dynam- two quadratures Xˆ and Pˆ of the field read out by the ics as a differential equation, similar to how physicists heterodyne measurement; the factor i between Lˆ and are accustomed to describing classical dynamics; further- X Lˆ is the 90◦ phase between these two orthogonal direc- more, the SME readily splits those dynamics into three P tions in the XP –plane (often also conventially labeled as terms, which make qualitatively distinct contributions to the IQ–plane). the dynamics. It is worth noting, however, that com- The resulting SME is then pared with the case of ordinary differential equations

[88], methods for the numerical integration of stochas- ρ˙ = Lˆ[ρ, LˆX ] + Lˆ[ρ, LˆP ] + Mˆ [ρ, LˆX ]ξX + Mˆ [ρ, LˆP ]ξP , tic differential equations [87] are more complex, and are (30) accurate only to substantially lower order in dt. Addi- where Lˆ and Mˆ are still the Lindblad dissipation, and tionally, direct numerical integration of the SME does measurement backaction terms, respectively. The Gaus- not necessarily preserve the properties of a valid density sian white noise for the measurement channels is char- matrix beyond O(dt), leading to problematic numerical acterized by each ξ(t) ∼ dW/dt. We may obtain equa- errors unless dt is extremely small; it is consequently nu- tions of motion in terms of Bloch sphere coordinates us- merically preferable to execute simulations of stochastic ing q˙ = tr(ˆσq ρ˙), yielding quantum trajectories by direct application of a positive mapping, as in (11) or similar, when possible. The in- q x˙ = − γ x + γ 1 + z − x2
ξ − x y ξ  , (31a) terested reader may find further comments in this vein 2 2 X P e.g. in Ref. [89]. γ q γ  2
 y˙ = − 2 y + 2 1 + z − y ξX − x y ξP , (31b) IV. SINGLE–QUBIT HETERODYNE TRAJECTORIES q γ z˙ = −γ(1 + z) − 2 (1 + z)[x ξX + y ξP ] , (31c) We now begin looking at diffusive quantum trajecto- in agreement with the result in eq. (25) of [66] (for u = 1+ ries due to heterodyne detection. What follows is essen- z, η = 1, and γφ = 0, in their notation). The stochastic tially a review of the simplest non–trivial case described readouts (signals arising from the measurement process) more extensively in Ref. [66], and corresponding to the are given by experimental implementation e.g. of Ref. [67]. In the q ˆ ˆ† γ language of quantum–limited amplifiers (QLAs), which rX = hLX + LX i + ξX = 2 x + ξX , (32a) are essential to realizing experiments involving individ- ual quantum trajectories, our meaning of “heterodyne” q corresponds to “phase–preserving” amplification (e.g. see ˆ ˆ† γ rP = hLP + LP i + ξP = 2 y + ξP . (32b) [28, 77–79] or similar, regarding implementations in cir- cuit QED scenarios). See Fig.1. Owing to the mixing Notice that the average path given by these equations of the fluorescence signal with a coherent state of the (where averages over an ensemble lead to ξ → 0, since field (the “local oscillator”, or LO), the heterodyne mea- these are zero–mean stochastic variables) obeys the same surement gives access to both quadratures of the field, basic fluorescence relations (10). This is a typical ex- with a symmetric uncertainty. A reader unfamiliar with ample of the relationship between an un-monitored and a quadrature phase space representation of a field mode continuously–monitored system, as we have discussed in may benefit from perusing e.g. Ref. [64]. When per- general above. formed with an ideal QLA, this scheme is formally equiv- We will interpret the equations (31) as being equations alent to projecting the field mode into the basis of the suitable for Itˆointegration and stochastic calculus (con- coherent states [66]. sistent with the assumptions used to derive (23) in the 10

2 first place [27]). It will also be useful to have the cor- such that ℘ = eC+Gdt+O(dt ). We see that up to the two responding Stratonovich version of this system of equa- last terms in Eq. (38), the probability density is Gaussian tions, which can be manipulated using regular calculus. in both readouts, with variances 1/dt, and means xpγ/2 In this case, the conversion (29) can be written as p and y γ/2 for rX and rP , respectively. Notice that this corresponds precisely to what we had from the SME, as A = a − 1 (b · ∇)b − 1 (b · ∇)b ; (33) 2 X X 2 P P in (32); the Gaussian form implicit in (38) is in fact key a trio of Stratonovich equations corresponding to the Itˆo in demonstrating that the form of the SME (23) written equations (31) are obtained by substituting this new drift in terms of Weiner increments dW is formally suitable vector (33) into (28). for this system. We also use the Kraus operator to obtain some equa- tions of motion. Consider the exapansion of the Kraus B. Kraus Operator Treatment operator itself to O(dt), which reads γ 2   ˆ |r| dt/4 − 2 0 2 We now consider the corresponding Kraus operator Mαe ≈ 11 + dt p γ +O(dt ) 2 (rX + irP ) 0 treatment of this situation. As discussed previously, a | {z } heterodyne measurement effectively projects the fluores- mˆ α cence signal onto a coherent state (A1) at each measure- (39) 2 2 2 ment timestep, such that we write down an operator for |r| = rX + rP . We can strip the Gaussian factor 2 e−|r| dt/4 from the operator for this purpose, since it ap-  √   √  1 −  0 2 1 −  0 pears in both the numerator and denominator of the state Mˆ = hα| √ |0i = e−|α| /2 √ . α aˆ† 1 α∗ 1 update expression (11), and thereby cancels off. Consider (34) the following series of approximations, assuming small dt:

We will use a substitution for the readouts given by † (11 +m ˆ αdt)ρ(t)(11 +m ˆ αdt) r ρ(t + dt) ≈ dt  †  α = (r − ir ); (35) tr (11 +m ˆ αdt)ρ(t)(11 +m ˆ αdt) 2 X P † †

≈ ρ + dt mˆ αρ + ρmˆ α − ρ tr mˆ αρ + ρmˆ α , the prefactor pdt/2 is chosen because it generates statis- (40) tics consistent with the shot–noise of the coherent state LO; for clarification see [66] and/or appendixA. With which can then be rearranged according to ρ(t + dt) − this substitution, we have a Kraus operator ρ(t) ≈ dt ρ˙, such that √ † †
  ρ˙ ≈ mˆ αρ + ρmˆ α − ρ tr mˆ αρ + ρmˆ α . (41) ˆ  dt 2 2  1 − γ dt 0 Mα = exp − 4 (rX + rP ) p γ , 2 dt(rX + irP ) 1 This can be expressed in Bloch coordinates by (36) γ q γ  2  x˙ = x z + rX (1 + z − x ) − rP x y , (42a) which may be used to update the state (using (11) 2 2 with Mˆ → Mˆ ) conditioned on acquiring a mea- r α q surement record drawn from the probability density y˙ = γ y z + γ r (1 + z − y2) − r x y , (42b)   2 2 P X ˆ ˆ † ℘(rX , rP |ρ(t)) = N tr Mαρ(t)Mα , where N is a nor- malization constant. The measurement operators form a γ 2 q γ proper POVM [55], in that z˙ = 2 (z − 1) − 2 (1 + z)[rX x + rP y] . (42c)

dt ZZ ∞ It is then straightforward to make the substitutions rX = dr dr Mˆ † Mˆ = 11, (37) p p 2π X P α α x γ/2 + ξX and rP = y γ/2 + ξP (32), and see that −∞ these equations from the Kraus operator approach are (i.e. the readouts we have defined here constitute another identical to the Stratonovich equations (28) obtained by complete set of measurement outcomes). conversion from the SME approach; this relationship be- It will be useful to take a closer look at the probability tween a Kraus operator based on Bayesian logic, and the density from which the readouts are drawn. Following SME, is consistent with previous results for this particu- the procedure we have typically used in the context of lar measurement [66], and other types of continuous qubit optimal paths (OPs) [32, 34, 47, 50, 66], we will expand measurements leading to diffusive SQTs [34, 35, 43, 47]. the log of the probability density to O(dt), defining a Simulations can be generated by applying the state up- ˆ ˆ term date rule (11) with Mr → Mα, with a pair of readouts drawn from Gaussians of means and variances described  q 2  q 2 G = − 1 r − γ x − 1 r − γ y above, at each timestep. The resulting stochastic tra- het 2 I 2 2 Q 2 (38) jectories diffuse as expected, and recreate the required γ 2 2
γ + 4 x + y − 2 (z + 1) decay dynamics on average, as shown in Fig.5(b). 11

(a) C. Generalizations

We consider the addition of a Rabi drive to the qubit (i.e. we now discuss additional tones inducing a unitary rotation in the Bloch sphere) by the addition of a Hamil- tonian term i[ρ, Hˆ ] to the SME (~ → 1), or a correspond- ing operator Uˆ = e−iHdtˆ to the measurement scheme with the Kraus operator (where the resulting equations of motion are insensitive to the order of operations, since they are only to O(dt)). Without loss of generality, we use Hˆ = δσˆz/2 + Ωˆσy/2, where we have denoted the de- 5 tuning δ = ωqb −ωdr, with ωdr the frequency of the tone . Such a tone induces a rotation around an axis tilted by (b) an angle arctan(Ω/δ) with respect to the z–axis. Note that the assumption in our derivation has been that only photons emitted by the qubit enter the transmission line which leads to the measurement apparatus; the simplest way to imagine engineering a system such that this re- mains valid with the Rabi drive on, is that the drive is being implemented by a tone which is off–resonant with the qubit/cavity/transmission line, such that the qubit photons couple to the output leading the measurement device only, and the drive photons couple to their own output only. As discussed earlier, this assumption also requires us to have the cavity resonance far from any of the Mollow triplet peaks, which√ are centered around ωdr 2 2 (c) and ωdr ± Ωeff, where Ωeff = Ω + δ is the generalized Rabi frequency; this regime and assumption is necessary if we want to treat the form of the decay channel as being unaffected by the drive. Drives of the type we have dis- cussed apply generically in “resonance fluoresence” sce- narios [6, 70, 90], as well as any other situation in which additional tones are present in qubit’s cavity (e.g. to im- plement additional measurements [73]). The situation we have described here is illustrated in Fig.1(c). Note that it is possible to generalize this heterodyne measurement by choosing the phase θ of the LO. The phase θ is a relative phase between the signal and LO, so it is equivalent to think of a phase plate having been FIG. 5. We show simulations of the decay from |ei to |gi under put in the signal line instead of the LO such thata ˆ† → ideal measurements, including photodetection (a), heterodyne e−iθaˆ† in (34), with interference against a fixed pump. detection for θ = 0 (b), and homodyne detection for θ = 0 Mathematically, we can then assign readouts according (c). In every case, we plot a dozen individual trajectories in to grey, the average trajectory over an ensemble of 10,000 sim- r ulated trajectories in solid blue, and the unmonitored curve dt α = eiθ(r − ir ), (43) integrated from (10) in dotted red. As required by the SME, 2 I Q we see good agreement between the simulated paths aver- aged over measurement noise realizations (solid blue), and a which leads to direct computation of the un–monitored dynamics, which in q γ the present case must simply follow z(t) = 2e−γt − 1 (dotted rI = 2 (x cos θ − y sin θ) + ξI , and (44) red). We can see the qualitative similarity between the dif- q γ fusive homodyne and heterodyne trajectories in (b) and (c), rQ = 2 (y cos θ + x sin θ) + ξQ. respectively, as well as their stark difference with the jump trajectories generated by photodetection (a); these contrasts are clear and important, as is the average dynamics common 5 to all three schemes, which follows from the shared underlying Note that this description is associated with a frame rotating at frequency ωdr, or equivalently the interaction picture with decay process at the heart of all three measurements consid- ˆ respect to Hframe = ωdrσˆz/2. In the fixed frame, the qubit ered here. All our simulations are performed by applying our ˆ Hamiltonian in presence of the drive reads H(t) = ωqbσˆz/2 + Kraus operator methods. iω t + −iω t Ω(iσˆ−e dr − iσˆ e dr ). 12

The operators for the SME which match the pair of ob- Then using a readout substituted in according to servables we infer from the means of G are r dt q q ˆ γ −iθ ˆ γ −iθ X → r, (48) LI = 2 e σˆ−, and LQ = i 2 e σˆ−. (45) 2 We see that changing the phase θ between the signal and we find that the POVM is normalized, i.e. LO effectively rotates the quadrature pair we measure. r Z ∞ dt ˆ † ˆ We have here used notation such that rI = rX and rQ = dr MxMx = 11. (49) 2π −∞ rP for the choice θ = 0. The relationship between the Kraus operator equations of motion and SME equations The relationship between X and r is again set based on of motion (Itˆoor Stratonovich) which we found in the comparing the readout statistics with LO’s shot noise, as θ = 0 case above, hold for arbitrary θ. discussed in appendixA. Those readout statistics can be We have reviewed the most basic features of an ide- readily understood from the expression alized heterodyne measurement. For a more advanced √ G = − 1 (r − γx)2 − γ (1 + z − x2), (50) treatment of this system, refer to [66]; we will now turn hom 2 2 our attention to applying the framework we have just which again comes from expanding the logarithm of developed to homodyne measurement. ˆ ˆ † tr(MxρMx). We infer that projecting onto |Xi in the photon space leads to a signal related to x in the qubit √ space, since r has a mean γx, and variance 1/dt. V. SINGLE–QUBIT HOMODYNE As before, we may takea ˆ† → aˆ†e−iθ to generalize the FLUORESCENCE TRAJECTORIES choice of measured quadrature, yielding an operator 1 √   4   Several experiments [69–72] and some theory [33] have dt 2 1 − γdt 0 Mˆ = e−r dt/4 √ , (51) been published about homodyne fluorescence measure- x 2π dt γ re−iθ 1 ment; we will develop our theory examples here far enough to compare them directly with the simplest ex- which still generates a proper POVM. Expanding the perimental results. log–probability density for the readout gives us √ Gθ = − 1 [r − γ(x cos θ − y sin θ)]2 hom 2 (52) γ 2 A. Kraus Operator and Measurement Dynamics − 2 (1 + z − (x cos θ − y sin θ) ); thus the mean of the Gaussian in r matches the signal Homodyne detection again involves interfering our sig- † √ given by Lˆ + Lˆ = γ(ˆσx cos θ − σˆy sin θ) for the SME nal with a strong LO. Practically, instead of amplify- √ operator Lˆ = γe−iθσˆ . ing both quadratures of the resulting signal as in hetero- − We proceed to find the equations of motion. Note that dyne detection, homodyne detection involves amplifying we can approximate Mˆ x as we did Mˆ α (39), such that one quadrature and de-amplifying the other [91] (“phase- √ sensitive” amplification). This procedure amounts to  1 − γdt 0   − γ 0  √ ≈ 11 + dt √ 2 +O(dt2). squeezing out the quadrature that isn’t measured, such dt γ re−iθ 1 γ re−iθ 0 that in the limit of ideal squeezing we project our signal | {z } onto a single quadrature’s eigenstate, instead of onto a mˆ x (53) coherent state [12, 24]. This yields a single readout sig- Then by the logic of (40) and (41) we may derive equa- nal, rather than the pair which arise in the heterodyne tions of motion in terms of Bloch coordinates case. We will follow the same recipe as in the heterodyne √ case, except that we project onto a final state |Xi (the γ  2  √ x˙ = 2 xz + Ωz − δy + r γ (1 + z − x ) cos θ + xy sin θ , eigenstate of the Xˆ = (ˆa†+ˆa)/ 2 operator in the quadra- (54a) ture space), instead of the coherent state |αi. Again, √ y˙ = γ yz + δx + r γ (y2 − z − 1) sin θ − xy cos θ , those unfamiliar with this phase space terminology may 2 (54b) wish to consult e.g. Ref. [64]. For dimensionless X, re- γ 2 √ call that we have the following solutions to the quantum z˙ = 2 (z −1)−Ωx+r γ(z +1) [y sin θ − x cos θ] . (54c) harmonic oscillator, which models the field mode: We have again used a Rabi drive characterized by Hˆ = 1 2 1 √ 2 − 4 −X /2 − 4 −X /2 hX|0i = π e , and hX|1i = π 2Xe . Ωˆσy/2 + δσˆz/2, or Uˆ ≈ 11 − iΩˆσydt/2 − iδσˆzdt/2. As (46) above, these equations are consistent with those derived √ −iθ Projecting onto the general fluorescence operator, and from the SME (23), using Lˆ = γe σˆ−, provided the 1 suppressing the factors π− 4 on all terms, we get SME output is correctly interpreted as an Itˆoequation, whose Stratonovich form then matches the above exactly.  √   √  1 −  0 2 1 −  0 Mˆ = hX| √ |0i = e−X /2 √ . Simulated trajectories for the case θ = 0 and Ω = 0 are x aˆ† 1 2X 1 shown in Fig.5(c), and demonstrate good agreement with (47) expectations, as in the previous cases. 13

B. Inefficient Measurements case discussed. Thus the description of η supposed by Fig.1(d) and (55) is entirely equivalent to the descrip- Inefficient measurement is easily included in the SME tion implicit in the SME, and clarifies the meaning of (23), and is completely described by the dimensionless measurement “inefficiency”. parameter η ∈ [0, 1]. In the Kraus operator picture, we This picture of inefficiency is also readily connected must modify the amplitude of the signal going into the to scenarios in which several observers simultaneously measurement apparatus; we will find a case intermediate make measurements, and each gets only partial infor- between perfect measurements (11) and no measurement mation [27, 92, 93]. One can imagine that an observer (20), reflecting that some fraction of the information is lives at each output of the beamsplitter in Fig.1(d), each lost rather than collected. A straightforward way to rep- recieving some proportion of the information about the resent this is with an unbalanced beamsplitter placed in qubit carried by the decay process as they make measure- front of our (still otherwise ideal) measurement device, as ments. If they do not share their results, each will have a shown in Fig.1(d). If ˆa† creates a photon in the emitted different estimate of the qubit’s evolution conditioned on field mode, the beamsplitter transforms it according to their partial information, and tracing out over the other observer’s measurement record which they do not have √ † † p † access to. Either of their estimates could be compared aˆ → η aˆs + 1 − η aˆ`, (55) to some hypothetical “true” evolution which an observer † where the surviving signala ˆs goes to the detector with able to access all the relevant measurement records could † compute. In practice, it is effectively impossible to have probability η, but the information in channela ˆ` is lost with probability 1 − η, and outcomes (all of which could a perfectly efficient measurement in any experiment, and have occurred) in the latter channel must be traced out. some information is always irretrievably lost to the en- We will do the trace of the lost channel in the Fock basis vironment through any channel from which the primary for simplicity (a sum of two terms is simpler than an inte- system is not perfectly isolated (generically, this is “de- gral over a continuous homodyne or heterodyne readout, coherence”). The methods we have presented here can although averaging over any complete set of hypotheti- readily be adapted to the kind of multiple–observer sit- cal measurement outcomes is technically correct). The uation we have just described; this includes situations scheme we are describing, for homodyne detection with which involve both jumps and diffusion, due to different efficiency η, can be implemented with a pair of operators observers making different types of measurements (see √ e.g. appendix B of [94], or [95]). Such scenarios have re-  1 −  0  cently been fruitfully investigated in the context of quan- Mˆ = hX j | √ |00i , (56) xj s ` † p † tum state smoothing [96–98]6. η aˆs + (1 − η)a ˆ` 1 We perform simulations which include measurement for Fock states j = 0, 1 in the lost mode, i.e. inefficiency, which are shown in Fig.7, and discussed fur-  √  ther in connection with the “optimal path” techniques −X2/2 1 −  0 Mˆ x0 = e √ , (57a) we develop shortly. Measurement inefficiency leads to 2ηX 1 decay that is qualitatively the same as in the ideal case discussed in Fig.5, except that instead of trajectories be-   ing restricted to pure states on the surface of the Bloch 2 0 0 Mˆ = e−X /2 , (57b) sphere, as in the η = 1 case, they instead move stochas- x1 p(1 − η) 0 tically on the surface of an ellipsoid which contracts to- with a state update rule wards |gi over time as information is lost in the η < 1 case. Qualitative agreement between these simulated re- ˆ ˆ † ˆ ˆ † sults shown in Fig.7 and those obtained in experiment for Mx0ρ(t)Mx0 + Mx1ρ(t)Mx1 ρ(t + dt) = . (58) either the homodyne [69] or heterodyne [67] detection can  ˆ ˆ † ˆ ˆ †  tr Mx0ρ(t)Mx0 + Mx1ρ(t)Mx1 be verified at a glance, and a quantitative understanding of this will be developed shortly. The measured homodyne signal is computed according As we have now successfully adapted and extended our to projection onto the states |Xi exactly as above, and a presentation of basic methods for pertaining to quantum drive could be added with unitaries in the same manner as above. The new operators (56) again denote a well– defined measurement, in that they form POVM elements, i.e. 6 Quantum state smoothing is closely related to quantum trajec- tories; SQTs, as we have presented them in the present text, are Z ∞ X ˆ † ˆ a form of “quantum filtering” which goes forward in time; in dX MxjMxj ∝ 11. (59) other words, we here only use the measurement record from the −∞ j=0,1 system’s past to estimate a qubit’s state. In the event of an inef- ficient measurement, quantum state smoothing often allows for We find the same agreement between the expansion of a more pure estimate of the system’s state to be made at some the state update (58) to O(dt), and the SME with fi- time, by using the measurement record both before and after the nite η (converted to its Stratonovich form), as in every time at which the state is estimated. 14 trajectories to the homodyne detection case, and estab- proach presented in Ref. [49], wherein similar and de- lished that they behave correctly, we can proceed by ex- tailed analysis was performed for the heterodyne case. tending our analysis of this system into new examples The time–reversed dynamics can be considered as a le- which can introduce and highlight particular topics in gitimate measurement dynamics, starting from the time– the recent literature. reversed final state Θ|ψf i, evolving through the time– reversed counterpart of the forward sequence of states, back to the time–reversed initial state Θ|ψii. The mea- VI. SPECIAL TOPICS AND FURTHER surement operators of the backward dynamics are related EXAMPLES to the forward dynamics by a Hermitian conjugate oper- ˆ ˆ † ation, i.e. MB = MF ; therefore the dynamical equations We will focus on connections to two areas, using ho- which describe the backward dynamics are also similar modyne fluorescence detection as our example of choice; to the retrodicted dynamical equations [105], but start- first we describe how this example relates to recent work ing from the time–reversed final state. We may write about the arrow of time in quantum measurement, which the retrodicted dynamical equations corresponding to a connects to work on fluctuation theorems for quantum homodyne measurement, where the quantum state is up- trajectories, and the growing area of quantum thermo- dated by dynamics more generally; second we will describe how Mˆ †ρ(t)Mˆ “most–likely paths” can be derived from diffusive quan- x ¯ x ρ(t + dt) = ¯† . (60) tum trajectory dynamics using a variational principle. ¯ ¯ ¯ tr[Mˆ xρ(t)Mˆ x] ¯ ¯ We have parameterized the single–qubit density matrix A. Time reversal symmetry and the arrow of time ρ with Bloch coordinates according to ¯ 1  1 + z x − iy  How does an arrow of time emerge from microscopi- ρ = ¯ ¯ ¯ . (61) ¯ 2 x + iy 1 − z cally time–reversible physical laws? The issue has been ¯ ¯ ¯ raised in the context of continuous quantum measure- Using the form of measurement operators given in ments [48, 49, 51, 93], and applies more broadly across Eq. (51) The dynamical equations now take the form, many disciplines within physics [99–101]. In the quan- tum measurement case, one could pose this question as γ √  2  a game; a quantum trajectory is shown like a movie, for- x˙ = 2 xz + r γ (1 − z − x ) cos θ + xy sin θ , (62a) ward and backward, and the goal of the game is to infer ¯ ¯¯ ¯ ¯ ¯¯ the direction in which the movie was originally recorded. √ y˙ = γ yz + r γ (y2 + z − 1) sin θ − xy cos θ , (62b) We will find that the equations of motion are time– ¯ 2 ¯¯ ¯ ¯ ¯¯ symmetric, (e.g. as in Hamiltonian dynamics), such that √ both the forward and backward movies both depict legit- z˙ = γ (z2 − 1) + r γ(−z + 1) −y sin θ + x cos θ . (62c) imate dynamics; this is not the whole story however, as ¯ 2 ¯ ¯ ¯ ¯ the backward evolution (i.e. “wavefunction uncollapse”) Note that the retrodicted equations under the time- does not necessarily occur with the same probability reversal operation, x → −x, y → −y, z → −z, and [102, 103]. This leads to a natural discriminator for the t → T − t (i.e., dt →¯ −dt) looks¯ exactly like¯ the forward arrow of time in terms of the probabilities of occurrence ¯dynamical equations¯ (54), demonstrating their time– of forward and backward trajectories of the monitored reversal invariance; we have eliminated the drive char- quantum system, as developed in Refs. [48, 49]. Assum- acterized by Ω and δ in the equations above for brevity, ing no prior bias, we could use the measurement record but including it does not affect the result. Time reversal as an additional tool to improve our inference about the symmetry of the dynamical equations suggests that the direction in which the quantum state dynamics is orig- forward dynamics and the reverse dynamics both repre- inally recorded [48] (by analogy, the sound track for a sent a physical quantum trajectory on the Bloch sphere. movie could help us understand in which direction it is Given the measurement record, one can associate a prob- meant to run). Such an approach is fundamentally con- ability each to the forward and backward trajectories, nected to the time–symmetry of underlying dynamical which can be used to infer an arrow of time for the mea- equations describing the measurement, and connects to surement dynamics, and subsequently characterize the the arrow of time analysis pertinent to the thermody- irreversibility of homodyne measurement of fluorescence namics of small systems [51, 104]. using the associated fluctuation theorems [48, 49, 51]. Fluorescence appears to exhibit a clear arrow of time, We can expand on this story somewhat by noting that and therefore the time–reversibility of continuously mon- our diffusive trajectories in Fig.5(b,c) do not diffuse itored fluorescence dynamics may seem rather surpris- monotonically downward from |ei towards |gi. This sug- ing. For this reason, here we detail the time symme- gests that the measurement process can actually cause try analysis of dynamical equations (54) which describe the probability of the qubit being found in the more en- homodyne measurement of fluorescence, using the ap- ergetic of its states to rise in some realizations. While 15 the average decays monotonically, fluctuations make this dan) path integral [32], and then apply the formalism to question of an arrow of time non–trivial in individual re- the homodyne fluorescence examples we have developed alizations; the probability of sustained re-excitation over above. a long period is low, but estimates of the arrow of time using only a short window of the evolution cannot nec- essarily be made with high confidence; rare events can 1. Derivation of Optimal Paths decieve, and something resembling a “wavefunction un- collapse” is not merely hypothetical in this system. Such OPs can be understood as the path extremizing the behavior has been noted in the literature [33]; while it probability to get from one given quantum state qi to an- may be initially intuitively challenging, this effect is per- other qf in a particular time interval, under the dynam- fectly correct, and reflects the nature of the information ics due to backaction from the continuous weak quantum we get about the field when we make a weak (dt  T1) measurement. The vector q parameterizes the quantum quadrature measurement, and its backaction. A truly state, and here denotes coordinates on the Bloch sphere. detailed description of the thermodynamics of quantum Typically OPs will be most–likely paths (MLPs), which measurements or trajectories falls beyond our present maximize the probability of the measurement record con- scope, but is a fascinating area related to the questions necting the given boundary conditions according to an we have discussed here, and enjoying increased recent re- action–extremization principle. OPs should be confused search interest [72, 106–113]. We encourage the curious neither with a globally most–likely path (i.e. the par- reader to explore further. ticular MLP post–selected on the most likely final state after a given time interval), or with an average path. De- tails about numerical procedures to extract an MLP from data, which corresponds to the theory we are about to B. A Variational Principle for Quantum develop, can be found in appendixB. Trajectories To begin, we explain how the path probability can be written in terms of an effective action, which can then Optimal paths (OPs) [32, 34, 47, 50, 66, 76] have re- be extremized according to a variational principle. We cently been used to elucidate a variety of quantum tra- may write an expression for the probability of a quantum jectory phenomena. We will here give a brief overview of trajectory, which moves from qi to qf through a discrete their derivation, using the CDJ (Chantasri/Dressel/Jor- sequence of measurements as

(n−1 ) Y P({q}, {r}|qi, qf ) = δ(qi − q0)δ(qf − qn) ℘(rk|qk)℘(qk+1|qk, rk) . (63) k=0 The δ–functions at the initial and final points impose the boundary conditions. The indices k run over time, such that if ρk = ρ(t), then ρ(t + dt) = ρk+1 and so on. The stochastic element of the dynamics arises in drawing the readout from the probability density determined by the denominator of the state update expression (11), i.e. ℘(rk|ρk) ∝ tr(Mˆ ρ Mˆ † ), where Mˆ could generically be any Kraus operator describing a weak measurement. We describe the rk k rk rk deterministic update of the quantum state given the stochastic readout rk according to ℘(qk+1|qk, rk) = δ(qk+1 − qk − dt F[qk, rk]), where q˙ = F[q, r] is an equation of motion, e.g. like (54). Recall that a δ–function may be written −dim(q) R i∞ δ(q) = (2πi) −i∞ dp exp [−p · q], where dim(q) is the dimension of q, and dp = dp1 dp2 ... dpdim(q). We apply this identity to all δ–functions in (63), such that in the time–continuum limit we have a Feynman–like path integral, in which we effectively sum over all possible quantum trajectories which obey the given boundary conditions, i.e.

i∞ n−1 ! n−1 Z Z Y  X  P = lim lim N ··· dpk exp B + (−pk · (qk+1 − qk − dt Fk) + ln ℘(rk|qk)) n→∞ dt→0 −i∞ k=0 k=0 Z " Z T # ∝ D[p] exp B + dt (−p · q˙ + p ·F[q, r] + G[q, r]) (64) 0 Z " Z T # Z = D[p] exp B + dt(H(q, p, r) − p · q˙ ) = D[p] exp (B + S[q, p, r]) 0

−(n+2)·dim(q) −dim(q ) for N = (2πi) , and where D[p] arises from the infinite product of the dpk(2πi) k . We use the shorthand B = −p−1 · (q0 − qi) − pn · (qn − qf ) for the boundary terms, and the shorthand G for the expansion to O(dt) of the log–probability for the readouts ln ℘(r|q) (see e.g. (52)). This relates a trajectory probability P to a “stochastic action” S. That action is expressed in terms of a Hamiltonian H = p ·F + G. 16

We can then see that extremizing the probability corresponds to extremizing S. Action extremization in this case can be expressed much the same way as it is in classical mechanics, such that

δS = 0 Z T Z T ∂H ∂H ∂H  = δ dt (−p · q˙ + H(q, p, r)) = dt · δq + · δp + · δr − p · δq˙ − q˙ · δp 0 0 ∂q ∂p ∂r (65) Z T       T ∂H ∂H ∂H = p · δq|0 + dt δq · + p˙ + δp · − q˙ + δr · , 0 ∂q ∂p ∂r

which indicates that the OPs obey the equations is imaginary). Continuing with this analogy, we could imagine that rays leave a source in different directions,

∂H ∂H ∂H i.e. with different wave–vectors (OPs leave their initial q˙ = , p˙ = − , = 0. (66) ∂p ∂q ∂r r? state qi with a range of initial momenta pi); we may pick out a particular ray or subset of rays by choosing a par- These are Hamilton’s usual equations in the Bloch coor- ticular initial wave–vector, or by choosing a final position dinates q and conjugate variables p, plus an additional to which they connect (a particular OP can be selected equation which stipulates that the stochastic readouts by choosing a value of pi, or by choosing a qf at a later ? be optimized, leading to some smooth r instead of the time). Thus, the degree of freedom in choosing an initial stochastic r. The OPs are thus smooth curves, and are momentum pi is the same degree of freedom which allows solutions to a Hamiltonian dynamical system of ordi- for post–selections to many qf ; the mapping between the nary, rather than stochastic, differential equations. The two is not necessarily one–to–one (see our work on “mul- OPs are themselves possible quantum trajectories (since tipaths” for clarification of this point [47, 50, 70]). In q˙ = F is preserved, by construction), even though the a phase space with N coordinates q and N momenta p ? optimal readouts r are smooth functions of time, rather (a 2N–dimensional phase space), the set of paths evolv- than the stochastic readouts which occur in individual ing from all pi with a fixed qi defines an N–dimensional runs of an experiment. The Hamiltonian structure of the Lagrangian manifold (LM) within the phase space; such OP dynamics implies that, absent any explicitly time– a manifold can be understood as containing all of the dependent parameters, there is a conserved “stochastic possible dynamics originating from the state qi in the energy” E = H associated with an OP. OP description, i.e. such an LM explores the full space The “momenta” p conjugate to the generalized Bloch of optimal readouts r?, just as the underlying diffusive coordinates q, which arise in this optimization process, process may explore the full space of stochastic measure- warrant further attention. The p are not directly measur- ment records8. Our focus now will be on applying this able, but play a substantive mathematical role, in that formalism to our homodyne fluorescence measurement; they effectively generate displacements in the quantum some work in this vein, albeit with a different emphasis, state q according to the optimal measurement record appears in Ref. [70]. r?. They appear as the variables conjugate to q in the Fourier representation of the δ–functions, and could be understood and Lagrange multipliers in the optimization process. They are perhaps best understood in analogy with classical optics, however: just as we have expressed 2. Optimal Paths for Homodyne Fluorescence Trajectories a diffusive process in terms of an action, the underly- ing wave process in classical optics can be expressed as an action; extremization in the latter scenario leads to a Notice that the system of equations (54) can be simpli- ray description. Our OPs are related to the underlying fied straightforwardly;y ˙ = 0 if y = 0 and θ = 0, and then diffusive process described above in much the same way all the dynamics are in the xz–plane of the Bloch sphere. that a ray description of light is related to the underly- ing wave optics process7. Diffusive SQTs arrive at final states with a variety of probabilities (and the action rep- resenting this is real), whereas optical paths leading to different positions arrive with different phases, exhibit- 8 The LM in question has primarily been used in the context ing interference (and the subsequent action in this case of multipath dynamics [47, 50, 70]; there the main concern is whether the projection of the time–evolved LM out of the full OP phase space down into the q–space of final quantum states is one–to–one (a single MLP connects the initial state to the cho- sen final state) or many–to–one (in which case many OPs may 7 We are grateful for comments by Prof. Miguel Alonso which connect the boundary conditions, typically corresponding to dif- helped us to clarify this point in our own thinking. ferent clusters of SQTs in the post–selected distribution). 17

It is then easy to write down a stochastic Hamiltonian p √ E (MHz) xz γ  2
Z- Hhom = px 2 xz + r γ 1 + z − x γ 2 √
+ pz 2 (z − 1) − r γ(z + 1)x (67) 1 √ 2 γ 2 − 2 (r − γx) − 2 (1 + z − x ), X+ X+ for the OP dynamics using formulas we have already de- Y- Y+ rived above (with Ω = 0 = δ, and η = 1) to describe the Y- 9 ideal measurement, its backaction, and statistics . The Y+ xz optimal readout obeys ∂rHhom|r? = 0, which we solve to obtain X-X-

? √ 2
r = γ x + px(1 + z − x ) − xpz(1 + z) ; (68) √ Z+ we see that we have the signal γx, plus some additional terms which depend on the conjugate momenta px and pz, which implement the optimized effect of the noise (as ϑ discussed above). We can simplify the equations even more. Consider a p change to polar coordinates according to the canonical S˙ (MHz) transformation Z- x → R sin ϑ z → R cos ϑ (69) px → pR sin ϑ + p cos ϑ/R X+ X+ p → p cos ϑ − p sin ϑ/R z R Y- Y+ which preserves the Poisson brackets between all of the Y- pairs of conjugate variables. Then we see that for the Y+ ˙ ?,Rϑ choice R = 1 and pR = 0, we have R = ∂pR Hhom = 0, meaning that we can look at dynamics purely on the great X-X- circle of the Bloch sphere at R = 1 (pure states), where states are parameterized entirely by a single coordinate Z+ ϑ (with ϑ = 0 ↔ |ei and ϑ = π ↔ |gi). Making this transformation, and subtituting in the optimal readout ? such that H = H|r=r? , we obtain the Hamiltonian ϑ  1 3γ  Hϑ? =p2 γ cos ϑ + γ cos(2ϑ) + hom 4 4 FIG. 6. We show the OP phase space in ϑ and p, for an ideal 3 1  homodyne measurement of the qubit fluorescence, and with + p γ sin ϑ + γ sin(2ϑ) (70) γ = 1 MHz and Ω = 0. We plot the stochastic Hamiltonian 2 2 in the top panel. Contours are lines of constant “stochastic 1 1 γ energy” E = Hϑ? in MHz, which are solutions to the OP − γ cos ϑ − γ cos(2ϑ) − , hom 2 4 4 dynamics. Fixed points appear at ϑ = 0 and π, at p = 0; the separatrices which pass through these points are shown in which generates the OPs in the simplest case we can con- magenta. Those separatrices bound off distinct regions X, Y, sider for this system. and Z; each has some distinct behavior, all of them ultimately The phase space for this Hamiltonian is plotted in lead their paths to the ground state θ = π. The regions come Fig.6, along with the time–derivative of the stochastic in pairs due to the symmetry of the phase space, where + action S which is extremized by the OP dynamics (ef- denotes that the paths in that region approach the ground fectively, S˙ gives an approximate representation of the state from below (with ϑ increasing), and their mirror images - approach from above (with ϑ decreasing). We plot S˙ = ϑ? ˙ Hhom −pϑ in the bottom panel; this quantity can be regarded as an approximate rate of probability decay, such that paths ˙ 9 Note that we have derived both F and G using regular calcu- which spend time in regions of more negative S correspond lus, and are thereby effectively using the Stratonovich form of to sequences of measurement results which are relatively less F, not the Itˆoform which arises directly from the SME (23). likely. The OP phase space regions are overlaid on the bottom Using a form of F which does not transform according to regu- plot in blue for reference. Comparing the two panels, we see lar calculus would prevent us from performing our OP analysis that we can associate paths with larger stochastic E with using typical approaches of classical mechanics (e.g. canonical less–likely dynamics (as a rule of thumb). transformations), which we find quite undesireable. 18 √ probability cost involved with traversing certain regions r = ηγ x + ξ, and the equations of motion from either of the OP phase space). A careful reading of these plots expanding (58) to O(dt), or converting the requisite SME can provide an insightful overview of the system dynam- from Itˆoto Stratonovich, are ics. First, we can immediately infer a rule of thumb: 2√ √ γ OPs with higher stochastic energy generically correspond x˙ = −rx γη+r(z+1) γη+ 2 x(η+ηz−1) = fx, (71a) to events which occur with lower probabilities (this is true to the extent that regions of large E correspond γ √
to regions of more–negative S˙). Secondly, we see that y˙ = y 2 (η + ηz − 1) − rx γη = fy, (71b) all paths in the OP phase space eventually approach |gi (ϑ = π) in the long–time limit, as we expect they must; γ √
there are possibilities for this to occur in either direction z˙ = (z + 1) 2 (η + ηz − 2) − rx γη = fz. (71c) around the Bloch sphere with some probabilities, but these pure–state OPs never cross through ϑ = π. The We immediately see that for this choice of measured quadrature, the y component of the dynamics can be uni–directionality of the flow towards |gi after t  T1 reflects our intuition that there should be a statistical eliminated with the choice y = 0, leaving only dynam- arrow of time in the measurement–induced dynamics, as ics in the xz–plane of the Bloch sphere; we will assume is discussed above and in detail elsewhere [49, 51]. A y = 0 for the remainder of this section. These assump- particular point in the phase space is worthy of further tions, with imperfect η, give us the simplest version of attention; the unstable fixed point at ϑ = 0 and p = 0 de- this system that can be compared directly with existing scribes an OP which is stationary at |ei for all time; this experiments. The last piece we need is an understanding does not violate our intuition however, since the proba- of the probability density function from which the read- bility cost involved with sitting at that point is greater outs are drawn; using the same methods as above, we than for sitting very close to |gi, such that it is still virtu- find ally impossible to post–select on a state still at |ei after η 1 √ 2 1 2
Ghom = − 2 (r − x ηγ) + 2 ηγ x − z − 1 . (72) t  T1. As noted before, it is possible for paths to start near the ground state, and re–excite, passing through Thus we see that simulations involve repeated state up- |ei before asymptotically approaching |gi again around dates as per (58), with readouts drawn at each step from √ the other side of the Bloch sphere; while such behavior a Gaussian of mean x ηγ and with variance dt−1. Op- corresponds to relatively rare events (a relatively low– timal paths are derived from a stochastic Hamiltonian probability post–selection is required), the possibility of such events is readily visible in the OP phase portrait10. η η H = pxfx + pzfz + G , (73) Further details about OPs this system, for the case of hom hom

Ω 6= 0, can be found in [70], and more detailed investiga- where fx and fz are the RHS of (71a) and (71c), re- tions of the corresponding heterodyne cases can be found spectively. Our aim below will be to elucidate the basic in [66]. aspects system dynamics, and show that our simulations and OPs match relevant results in the experimental lit- erature. C. Optimal Paths for Inefficient Measurements, A particularly important feature of the dynamics un- and Connections to Experiments der homodyne fluorescence detection (absent a Rabi drive or other dynamics) is that all trajectories are constrained We develop a final example; in order to compare OPs to an ellipse in the Bloch sphere at any given time [69, 71] directly to experimental results, we need to introduce (and a similar ellipsoid is apparent in the heterodyne case measurement inefficiency. In the homodyne case with [68]). The functional form of these ellipses has been de- θ = 0 and Ω = 0 = δ, corresponding to the SME readout rived in the literature [71], and follows 1 h i z(t) = 1 ± p1 − u(t)x2 − 1, (74) u(t) 10 For example, starting at ϑ0 = π − 1, a globally more–likely path which we would regard as typical might arise from post–selecting for the time–dependent function on ϑ = π − 0.01 (≈ ground) at some later time, picking out a f γt solution from X+. However, it is possible to post–select on some u(t) = η + (u0 − η)e , (75) state like ϑf = −π + 0.01 (also ≈ ground) instead, selecting a path from X- or Y-; this reveals the possibility of a much rarer where u0 is set by the initial state according to set of events, corresponding to an OP which circles back through the excited state before decaying towards ground, from the op- 2 x2 posite direction as compared with the more typical set of paths. 0 u0 = − 2 . (76) Post–selections drawing out these dynamics can generically se- 1 + z0 (1 + z0) lect OPs from regions X or Y of the phase portrait, as detailed in Fig.6; those in region Z can only partially re-excite, before For example, with the initial state |ei, we have u0 = 1 turning around to decay. at t = 0, and at any time t > 0 all possible trajectories 19

z (a) z (b) t = T1/2 t = T1 (e)

E [γ]

t = T1/2

x x t = T1

z (c) z (d) t = 3T1/2 t = 3T1/2 t = 2T1

t = 2T1 etc.

x x

FIG. 7. We show the evolution of the qubit state from |ei, under the dynamics described by (71) and (73), for a realistic measurement efficiency η = 0.45. We show the density of simulated quantum trajectories at different times in (a–d); our methods reproduce the ellipses (74), demonstrating quantitative agreement with the relevant experimental literature [69, 71]. In (e), we plot the Lagrangian manifold projected down into the xz Bloch plane from the OP phase space at different times, starting with t = T1/2, up to t = 4T1 in increments of T1/2. The known analytic solutions (74) are plotted in grey, such that we can see the exact correspondence between the sampled LM, literature, and our simulations. The ellipsoids here and in Fig.3 bear obvious similarities; we stress however, that there are several qualitative differences between these two cases. In the case of Fig.3 we lose all the information, and many initial states, which originate deterministic solutions to (10), are required to see the decaying ellipses; in the present case, the ellipsoids arise from a single initial state, due to different measurement records which contain partial information from the qubit’s optical environment. evolving from |ei under dynamics from the inefficient ho- which lives in the four–dimensional phase space, into a modyne measurement can be found on the ellipse (74) plot that appears in the coordinates x and z only, at se- (as a function of x); this ellipse is initially the great circle lected times). We then see that we have exact agreement bounding the xz–plane of the Bloch sphere, and decays between the LM and the analytic curves (74), consistent towards the ground state according to the time depen- with the fact that the OPs are themselves possible quan- dence (75). We develop this example in Fig.7. In the left tum trajectories. This reinforces our statements about four panels (a–d) we show the density of simulated trajec- the consistency between the methods reviewed here, and tories originating at |ei after different evolution times; we the broader literature on continuous monitoring of fluo- find essentially perfect agreement between the histograms rescence, but also serves to illustrate the role of the initial of these simulated trajectory densities, and the analytic momenta pi in the OP formalism. Choosing a particu- curves (74) known from the literature. We stress that in lar pi selects particular boundary conditions from the departure from many other quantum measurement sce- possible multitude, and the complete set of pi contained narios, there are final states on which it is impossible to in the LM index a complete set of possibilities for the post–select in the present system; typically some states OPs originating at a particular state. While the LM in appear very rarely in the dynamics, but here large regions question may at first seem a somewhat abstract mathe- of the Bloch sphere are forbidden entirely. matical object, we are here able to highlight its physical We conclude by demonstrating the connection between character. the Lagrangian manifold from the OP phase space we described above, and the ellipses we have just described. The relative simplicity of the dynamics under inefficient VII. CLOSING REMARKS homodyne fluorescence measurement make this an ideal example with which to illustrate the concepts discussed We have given an overview of many useful methods and above. In Fig.7(e) we show the projection of the LM insights that arise from considering continuous quantum originating at |ei into the xz–plane of the Bloch sphere measurement, emphasizing examples in which we track (i.e. we evolve the OP equations sampled across the ini- a quantum emitter’s state by gathering and measuring tial LM, and then flatten the two–dimensional manifold, spontaneously–emitted photons. We have focused on a 20

Kraus operator approach to this problem, most similar emission; none of the other details of our photodetection to that developed in Ref. [66], and made connections to story carry over in a simple way, however, to the case of a corresponding stochastic master equation description generalized measurements. All of this is related to the throughout. Many of the issues which arise in treating fact that the emission process entangles the qubit with this particular type of system are common to stochas- the field mode (see e.g. the transition from (12) to (13)); tic quantum trajectories in general, and we have conse- this correlation allows us, as observers, to infer an emitter quently addressed many of the important principles and state based on the field, but also means that the type of typical problems one needs to become aware of when en- information we infer about the emitter depends on what tering this research area. We have also been able to use kind of state we project the field into (we must choose the fluorescence examples above to offer accessible illus- what kind of question to ask the field, and this will affect trations and introductions to selected advanced topics of the kind of answer we can subsequently expect). Such contemporary interest; for example, we have been able issues get to the heart of quantum mechanics, and quan- to make comments to help the interested reader engage tum trajectories generally serve as an excellent point of with work on the arrow of time in quantum trajectories, departure for such discussions about the foundations and or understand how to generate and interpret trajectories interpretations of the theory, and the role of an observer which follow an optimal measurement record. probing a system which is otherwise sufficiently isolated We can take a larger view of the processes we have so as to behave “quantumly”. described. We are accustomed to talking about the flu- orescence process in terms of the emission of individual photons at particular times, from a sudden jump in the There are, of course, many related topics which we qubit state, i.e. we typically discuss fluorescence in lan- have not been able to cover at all, but which we hope guage which lends itself naturally to the photodetection may become substantially easier for a new reader to di- case shown in Fig.5(a). This notion of a jump and pho- gest with the foundation we have developed above. We ton emerging at a particular time were mathematically highlight three in particular, in addition to those of a enforced in the framework presented above by choosing thermodynamic character [72, 107, 108, 110], pertaining outcomes in the Fock basis of the field, and with perfect to quantum state smoothing [40, 96–98], or the optimal measurement efficiency, we can say that we have collected path methods [32, 34, 47, 50, 66, 76] we have already complete information about the output mode. The other mentioned. First, some of the main interest in quantum measurements we have discussed, are however, just as trajectories is geared towards its applications to quan- “complete” (in the sense that we have a POVM, and can tum control; we point out some literature which adapts ascribe a pure state to the qubit at any time in the η = 1 the types of measurements we have described above to case; even in the η < 1 case, we are able to assign a state this purpose [68, 114]. Second, dispersive measurements to the qubit which is consistent with a subsequent tomo- [43–45, 115], allowing for measurements of e.g. Lˆ ∝ σ , graphic verification [68, 69]). The trajectories generated z are extremely common in the literature; unlike the mea- by the heterodyne and homodyne measurement schemes surements we have described here, a qubit in a cavity (or do not readily admit interpretation in terms of a photon coupled to a resonator) is directly probed with a pulse emerging at any particular time; there is no single point which reveals information about its state; virtually ev- along a trajectory in Fig.5(b or c) to which we can point erything in terms of general approach we have developed and say “this is when the photon was emitted”. However, above can be carried over to this case however. Such we can still ascribe a stochastic evolution to the qubit measurements have also been utilized in contemporary state, which agrees with any sensible check we know how feedback control problems [52–54], or in simultaneously to perform, and that evolution evidently depends on how weakly measuring multiple non–commuting observables or what type of environmental information was collected; [46, 47, 50, 116]. Combinations of dispersive and fluores- the dynamics on average reflect the same decay statistics cence measurements have been realized experimentally we are used to regardless of the measurements (which is [73]. An introduction to a simple model of dispersive always the case in such continuous quantum measure- measurements, similar in spirit to that above, can be ment problems, and more or less the only assumption we found in the appendix of Ref. [47] and references therein. made to formulate the model we have used). In other Third, direct extensions of the measurements we have dis- words, we the observer can dynamically assign our qubit cussed here to the two–qubit case can serve as a spring- a state11 which may e.g. begin at |ei and eventually wind board to study measurement–induced entanglement gen- up in |gi according to the typical statistics of spontaneous eration between a pair of emitters [94, 117–119], and the decay of entangled states open to the influence of de- cay channels and/or measurements [120–124]. “Bell state measurements” are essential in recent tests of quantum 11 One could say that we, the observer, are updating our proba- balistic prediction about outcomes of a future measurement; this mechanics and local realism [125]. As such, we hope that “best guess” about possible future outcomes is made based on our present work may help new readers to better digest a some preparation procedure, past measurement record, and the wide variety of literature concerning quantum measure- rules we typically refer to as “quantum mechanics”. ments, open quantum systems, and beyond. 21

ACKNOWLEDGMENTS We can talk about the fluctuations either in photon num- ber, or LO power, which are We acknowledge funding from NSF grant no. DMR- r dt hP i 1809343, and US Army Research Office grant ∆N = phNi = , or (A5a) no. W911NF-18-10178. PL acknowledges additional hν support from the US Department of Education grant No. GR506598 as a GAANN fellow, and thanks the r hν hν hP i Quantum Information Machines school at Ecole´ de ∆P = phNi = (A5b) Physique des Houches for their hospitality during part dt dt of this manuscript’s preparation. We are grateful to Thus we stress that for fixed ν and hP i, the fluctuations Joe Murphree for helpful comments. Analyses of the √ in the LO photon number go like dt, and the fluctu- type described in appendixB and Fig.8 have benefited √ ations in the LO power go like 1/ dt. In other words, from numerous conversations with Areeya Chantasri, for fixed average LO power, there are fluctuations in the and some versions of the underlying numerical methods subsequent photocurrent with variance 1/dt, and it is have benefited from work by John Steinmetz and Kurt precisely these physics which motivate the assignments C. Cylke. Numerical methods underlying Figs.3,5,7, (35) and (48) in modeling the heterodyne and homodyne and8 have been implemented in Python 2.7. measurements, respectively. Those fluctuations are well approximated as Gaussian for large hNi (the Poisson dis- tribution converges to a Gaussian for large numbers). APPENDICES That our formalism leads to agreement with standard tools like the SME, and with experimental data (e.g. as Appendix A: LO Power and Photocurrent in Fig.7), further justifies the use of the intuition we pro- Fluctuations vide here, connecting our measurement noise to the shot noise of the measurement LO / amplification pump. We consider the logic behind the scaling of the read- outs in time in the dyne measurements, i.e. we justify the expressions (35) and (48). The argument we offer follows Appendix B: Connection Between Optimal Paths directly from Ref. [66]. Dyne measurements involve in- and Simulation / Data terfering the signal beam (which here contains only zero or one photon) with a strong coherent state LO. Recall Our expectation in deriving most–likely paths (MLPs) that a coherent state is that, in typical cases, they should correspond approxi- mately to a highest–probability peak in the post–selected ∞ n 2 X α sub–ensemble of trajectories connecting an initial and fi- |αi = e−|α| /2 √ |ni (A1) n! nal state (see comments below about some of the sub- n=0 tleties, however). Our aim here is to describe the proce- dure by which we approximate this concept in extracting has a mean photon number a MLP from simulated SQTs or data, and show by ex- ample that such results are in good correspondence with hNi = hα|a†a|αi = |α|2 (A2) those given by the CDJ optimization (see Sec. VI B 1). This has been performed in a variety of cases elsewhere with fluctuations [45, 50, 70], and we here describe and perform the requi- q 2 p site analysis for solutions of (70) and the corresponding ∆N = ha†aa†ai − ha†ai = |α| = hNi. (A3) simulation. We generically suppose that we are given a simulated This is a direct result of the fact that a coherent state ensemble of SQTs {ρ(t)}, initialized at a particular ρ0; generates the Poisson statistics for photon arrival times; such trajectories are necessarily sampled over some small this is typical for arrival times pertaining to any ran- but discreet timestep dt, consistent with the simulation dom process characterized by a constant average rate. A procedures described in the main body of the text above. constant underlying rate is, of course, consistent with an We describe the numerical manipulations we perform on assumption that the LO has constant power (on aver- such a set of {ρ(t)} to extract an “experimental MLP” age, up to the quantum fluctuations). If hν is the energy which may be compared directly with theory. These go per photon at frequency ν in the LO beam, then the LO as follows: power corresponds to the average photon number arriv- ing in a time interval dt according to hP i = hNi hν/dt, 1. We begin by imposing the final boundary condi- p and we have fluctuations hNi ± hNi or tion, i.e. we post–select on the desired ρT , at a later time T . This means that we must pick a dis- hν   tance measure D(ρ1, ρ2) between quantum states hP i ± ∆P = hNi ± phNi . (A4) dt (e.g. fidelity, Bures distance, or similar), and keep 22

the sub–ensemble for which D(ρ(t = T ), ρT ) ≤ W , where W is some small widow or allowed tolerance about the chosen final state. We’ll call the post– selected sub–ensemble (PSSE) {ρ(t)}ps; this is, by construction, the set of trajectories which connect (a) ρ0 to ρT . ϑ –|ei 2. A simple intuition about the meaning of the MLP is that it should follow a densest cluster of trajectories in {ρ(t)}ps; in order to approximate this concept of a “densest cluster” numerically, we must rank each SQT in {ρ(t)}ps according to its distance to all –|gi other SQTs in {ρ(t)}ps. It is useful to construct a matrix of elements Dnm, where n and m are indices – which run over the SQTs in {ρ(t)}ps; we write (b) |gi X Dnm = D(ρn,k, ρm,k), (B1) k

where the sum over k runs over all the timesteps between t = 0 and t = T . The matrix D will ϑ –|ei be symmetric as long as the distance measure D is symmetric (we strongly discourage the use of any asymmetric distance measures for our present purposes). Then each SQT can be assigned a dis- tance score relative to all other elements of {ρ(t)}ps ¯ P according to Dn = Dnm, where a relatively |gi ¯ m – smaller value of Dn indicates that trajectory n of is closer to other trajectories in the post–selected set. These distances scores thus allow us to rank all of the trajectories in the PSSE. FIG. 8. We compare ideal (pure–state) simulations against OPs generated by (70), between a few boundary conditions. 3. The final step in the procedure is a simple average; All times (x–axes) are expressed in units of T1, and all cases we take the the closest–clustered 5%-10% of trajec- shown use Ω = 0. We plot the density of trajectories initial- tories in {ρ(t)}ps, (those with the smallest 5%-10% ized at a given state ϑi, and post–selected to within ϑf ± 0.01 of D¯ ), and average them. The idea is that this ap- at the final time on each plot; the density scale, which is n normalized between the initial and final timesteps, is shown proximates the smooth curve following the densest on the accompanying colorbar. The “experimental MLP” ex- cluster of SQTs in {ρ(t)}ps. tracted from the post–selected SQTs, as described in sec.B, is shown over the density in dotted black, while the correspond- We apply this procedure, and compare with the analytic ing curve derived from the stochastic Hamiltonian (70) is solutions to (70), for a few selected boundary conditions, 1 in Fig.8. shown in solid blue. In (a), we use ϑi = 2 , and ϑf = −π+0.01 at T = 10.1T1 (the post–selection window keeps SQTs which We close with a few remarks about the procedure we satisfy ϑ(10.1) ∈ [−π, −π+0.02]). These boundary conditions have just described. We lack a formal proof that the nu- are chosen such that the MLP shown approximately follows merical procedure just outlined necessarily always con- the separatrix in Fig.6 from a state near |ei, over the ex- verges to the optimization we perform by the CDJ path cited state, before decaying towards |gi. In (b), we look at a integral method [32]. We nonetheless see in Fig.8 that somewhat less likely case, where trajectories are initialized at the present case continues to support the agreement be- ϑi = 0 (the excited state |ei), and the post–selection is applied 1 tween SQTs (from either simulation, or real data) and at ϑf = −π + 2 at T = 3T1; so we are looking at anomalously slow decay, which is still hovering a bit above the ground state OPs, which have been successfully compared in many |gi after several decay times. We note that we have excellent other scenarios as well [45, 50, 70]. An important feature agreement between the theoretical and numerical MLPs. It is of the numerical procedure we have described is that it also apparent that paths which dip “below” our chosen ϑf and ranks trajectories by considering their entire evolution then rise back to it are exceedingly rare; there are far more connecting qi and qf ±˜ W , rather than looking for some events in this PSSE which decayed partially to the “wrong” piecewise optimization, or explicitly following a peak tra- side of the Bloch sphere at early times and then came back jectory density at particular times. This reflects the char- over |ei to meet the final boundary condition. acter of the variational approach we have used in the theory; invoking δS = 0 leads to an optimization over an entire trajectory duration. While in many simple cases 23 some piecewise numerical optimization could get us sim- step 3 above. The number of trajectories before post– ilar results, we would expect this simplified picture to selection, the size of the post–selection window, and the fail in other cases, as it makes a substantial conceptual overall probability for trajectories to reach the desired departure from the theory we want these numerics to final state, all play a role in these final numbers. The match. In terms of practical considerations, we point finer the post–selection window, the more initial trajec- out that the “experimental MLP” procedure above leads tories are required at the beginning in order to obtain to an attrition in the number of SQTs used at each step, smooth results. Post–selections on very rare events may first due to post–selection, and then due to the ranking be prohibitively difficult to verify in practice, simply due procedure. In order to obtain smooth results, it is neces- to the overwhelming amount of data that would need to sary that at least several hundred SQTs make it into the be collected / generated in order to have an appropriate final, most–closely clustered group which are averaged in number of SQTs in the PSSE.

[1] Albert Einstein, “Strahlungs-Emission und Absorp- A: Mathematical and General 25, 5165–5176 (1992). tion nach der Quantentheorie,” Deutsche Physikalische [20] N. Gisin and I. C. Percival, “The quantum-state diffu- Gesellschaft 18 (1916). sion model applied to open systems,” Journal of Physics [2] Paul A. M. Dirac, “The quantum theory of the emission A: Mathematical and General 25, 5677–5691 (1992). and absorption of radiation,” Proceedings of the Royal [21] H. M. Wiseman and G. J. Milburn, “Quantum theory of Society A 114 (1927). optical feedback via homodyne detection,” Phys. Rev. [3] V. Weisskopf and E. Wigner, “Berechnung der Lett. 70, 548–551 (1993). nat¨urlichen linienbreite auf grund der diracschen licht- [22] H. M. Wiseman and G. J. Milburn, “Quantum theory theorie,” Zeitschrift f¨urPhysik 63, 54–73 (1930). of field-quadrature measurements,” Phys. Rev. A 47, [4] B. R. Mollow, “Power spectrum of light scattered by 642–662 (1993). two-level systems,” Phys. Rev. 188, 1969–1975 (1969). [23] H. M. Wiseman and G. J. Milburn, “Interpretation of [5] Jay R. Ackerhalt, Peter L. Knight, and Joseph H. quantum jump and diffusion processes illustrated on the Eberly, “Radiation reaction and radiative frequency bloch sphere,” Phys. Rev. A 47, 1652–1666 (1993). shifts,” Phys. Rev. Lett. 30, 456–460 (1973). [24] H. M. Wiseman, “Quantum trajectories and quantum [6] H. J. Kimble and L. Mandel, “Theory of resonance flu- measurement theory,” Quantum and Semiclassical Op- orescence,” Phys. Rev. A 13, 2123–2144 (1976). tics: Journal of the European Optical Society Part B 8, [7] P. W. Milonni, “Why spontaneous emission?” American 205–222 (1996). Journal of Physics 52, 340–343 (1984). [25] Todd A. Brun, “A simple model of quantum trajecto- [8] Serge Haroche and Daniel Kleppner, “Cavity quantum ries,” American Journal of Physics 70, 719 (2002). electrodynamics,” Physics Today 42, 24 (1989). [26] John Gough and Andrei Sobolev, “Continuous mea- [9] H. J. Carmichael, An Open Systems Approach to Quan- surement of canonical observables and limit stochas- tum Optics (Springer, Berlin, 1993). tic Schr¨odingerequations,” Phys. Rev. A 69, 032107 [10] C. W. Gardiner and P. Zoller, Quantum Noise: A (2004). Handbook of Markovian and Non-Markovian Quantum [27] Kurt Jacobs and Daniel A. Steck, “A straightforward in- Stochastic Methods with Applications to Quantum Op- troduction to continuous quantum measurement,” Con- tics (Springer, 2004). temporary Physics 47, 279–303 (2006). [11] I. C. Percival, Quantum State Diffusion (Cambridge [28] A. A. Clerk, M. H. Devoret, S. M. Girvin, Florian Mar- University Press, 1998). quardt, and R. J. Schoelkopf, “Introduction to quan- [12] H. M. Wiseman and G. J. Milburn, Quantum measure- tum noise, measurement, and amplification,” Rev. Mod. ment and control (Cambridge University Press, 2010). Phys. 82, 1155–1208 (2010). [13] A. Barchielli and M. Gregoratti, Quantum trajectories [29] M. F. Santos and A. R. R. Carvalho, “Observing differ- and measurements in continuous time (Springer-Verlag ent quantum trajectories in cavity QED,” EPL (Euro- Berlin Heidelberg, 2009). physics Letters) 94, 64003 (2011). [14] Kurt Jacobs, Quantum Measurement Theory and its [30] John E. Gough, Matthew R. James, Hendra I. Nurdin, Applications (Cambridge University Press, 2014). and Joshua Combes, “Quantum filtering for systems [15] N. Gisin, “Quantum measurements and stochastic pro- driven by fields in single-photon states or superposition cesses,” Phys. Rev. Lett. 52, 1657–1660 (1984). of coherent states,” Phys. Rev. A 86, 043819 (2012). [16] Alberto Barchielli, “Measurement theory and stochas- [31] Andrew N. Jordan, “Quantum physics: Watching the tic differential equations in quantum mechanics,” Phys. wavefunction collapse,” Nature 502, 177–178 (2013). Rev. A 34, 1642–1649 (1986). [32] A. Chantasri, J. Dressel, and A. N. Jordan, “Action [17] Carlton M. Caves and G. J. Milburn, “Quantum- principle for continuous quantum measurement,” Phys. mechanical model for continuous position measure- Rev. A 88, 042110 (2013). ments,” Phys. Rev. A 36, 5543–5555 (1987). [33] Anders Bolund and Klaus Mølmer, “Stochastic excita- [18] L. Di´osi,“Continuous quantum measurement and Itˆo tion during the decay of a two-level emitter subject to formalism,” Physics Letters A 129, 419 – 423 (1988). homodyne and heterodyne detection,” Phys. Rev. A 89, [19] N. Gisin and M. B. Cibils, “Quantum diffusions, quan- 023827 (2014). tum dissipation and spin relaxation,” Journal of Physics [34] Areeya Chantasri and Andrew N. Jordan, “Stochastic 24

path-integral formalism for continuous quantum mea- ous quantum measurements and absolute irreversibil- surement,” Phys. Rev. A 92, 032125 (2015). ity,” Phys. Rev. A 99, 022117 (2019). [35] Alexander N. Korotkov, “Quantum Bayesian approach [52] Taylor Lee Patti, Areeya Chantasri, Luis Pedro Garc´ıa- to circuit QED measurement with moderate band- Pintos, Andrew N. Jordan, and Justin Dressel, “Lin- width,” Phys. Rev. A 94, 042326 (2016). ear feedback stabilization of a dispersively monitored [36] Jonathan A. Gross, Carlton M. Caves, Gerard J. Mil- qubit,” Phys. Rev. A 96, 022311 (2017). burn, and Joshua Combes, “Qubit models of weak [53] S. Hacohen-Gourgy, L. P. Garc´ıa-Pintos, L. S. Martin, continuous measurements: Markovian conditional and J. Dressel, and I. Siddiqi, “Incoherent qubit control open-system dynamics,” Quantum Science and Technol- using the quantum Zeno effect,” Phys. Rev. Lett. 120, ogy 3, 024005 (2018). 020505 (2018). [37] John E. Gough, “An introduction to quantum filtering,” [54] Z. K. Minev, S. O. Mundhada, S. Shankar, P. Reinhold, arXiv:1804.09086 (2018). R. Gutierrez-Jauregui, R. J. Schoelkopf, M. Mirrahimi, [38] John E. Gough, “The Gisin-Percival stochastic H. J. Carmichael, and M. H. Devoret, “To catch and Schr¨odingerequation from standard quantum filtering reverse a quantum jump mid-flight,” Nature 570, 200 theory,” Journal of Mathematical Physics 59, 043509 (2019). (2018). [55] M. A. Nielsen and I. L. Chuang, Quantum Computa- [39] John E. Gough, “How to estimate past measurement tion and Quantum Information (Cambridge University interventions on a quantum system undergoing contin- Press, 2000). uous monitoring,” arXiv:1904.06364 (2019). [56] J. von Neumann, Mathematical Foundations of Quan- [40] John E. Gough, “The collapse before a quantum jump tum Mechanics (Springer, 1932). transition,” Reports on Mathematical Physics 85, 69 – [57] Serge Haroche and Jean-Michel Raimond, Exploring the 76 (2020). Quantum: Atoms, Cavities, and Photons (Oxford Grad- [41] Gavin Crowder, Howard Carmichael, and Stephen uate Texts) (Oxford University Press, 2006). Hughes, “Quantum trajectory theory of few-photon [58] Yakir Aharonov, David Z. Albert, and Lev Vaidman, cavity-qed systems with a time-delayed coherent feed- “How the result of a measurement of a component of back,” Phys. Rev. A 101, 023807 (2020). the spin of a spin-1/2 particle can turn out to be 100,” [42] N. Gisin, P. L. Knight, I. C. Percival, R. C. Thompson, Phys. Rev. Lett. 60, 1351–1354 (1988). and D. C. Wilson, “Quantum state diffusion theory and [59] Boaz Tamir and Eliahu Cohen, “Introduction to weak a quantum jump experiment,” 40, 1663–1671 (1993). measurements and weak values,” Quanta 2, 7–17 (2013). [43] Jay Gambetta, Alexandre Blais, M. Boissonneault, [60] Christopher A. Fuchs and Asher Peres, “Quantum-state A. A. Houck, D. I. Schuster, and S. M. Girvin, “Quan- disturbance versus information gain: Uncertainty rela- tum trajectory approach to circuit QED: Quantum tions for quantum information,” Phys. Rev. A 53, 2038– jumps and the Zeno effect,” Phys. Rev. A 77, 012112 2045 (1996). (2008). [61] Christopher A. Fuchs and Kurt Jacobs, “Information- [44] K. W. Murch, S. J. Weber, C. Macklin, and I. Siddiqi, tradeoff relations for finite-strength quantum measure- “Observing single quantum trajectories of a supercon- ments,” Phys. Rev. A 63, 062305 (2001). ducting quantum bit,” Nature 502, 211 (2013). [62] Ognyan Oreshkov and Todd A. Brun, “Weak mea- [45] S. J. Weber, A. Chantasri, J. Dressel, A. N. Jordan, surements are universal,” Phys. Rev. Lett. 95, 110409 K. W. Murch, and I. Siddiqi, “Mapping the optimal (2005). route between two quantum states,” Nature 511, 570 [63] Gerd Leuchs, “Squeezing the quantum fluctuations of (2014). light,” Contemporary Physics 29, 299–314 (1988). [46] S. Hacohen-Gourgy, L. S. Martin, E. Flurin, V. V. Ra- [64] Christine Silberhorn, “Detecting quantum light,” Con- masesh, K. B. Whaley, and I. Siddiqi, “Dynamics of temporary Physics 48, 143–156 (2007). simultaneously measured non-commuting observables,” [65] P. Campagne-Ibarcq, L. Bretheau, E. Flurin, Nature 538, 491 (2016). A. Auff`eves, F. Mallet, and B. Huard, “Observ- [47] Philippe Lewalle, Areeya Chantasri, and Andrew N. ing interferences between past and future quantum Jordan, “Prediction and characterization of multiple ex- states in resonance fluorescence,” Phys. Rev. Lett. 112, tremal paths in continuously monitored qubits,” Phys. 180402 (2014). Rev. A 95, 042126 (2017). [66] Andrew N. Jordan, Areeya Chantasri, Pierre Rouchon, [48] Justin Dressel, Areeya Chantasri, Andrew N. Jordan, and Benjamin Huard, “Anatomy of fluorescence: Quan- and Alexander N. Korotkov, “Arrow of time for con- tum trajectory statistics from continuously measuring tinuous quantum measurement,” Phys. Rev. Lett. 119, spontaneous emission,” Quantum Studies: Math. and 220507 (2017). Found. 3, 237 (2015). [49] Sreenath K. Manikandan and Andrew N. Jordan, “Time [67] P. Campagne-Ibarcq, P. Six, L. Bretheau, A. Sarlette, reversal symmetry of generalized quantum measure- M. Mirrahimi, P. Rouchon, and B. Huard, “Observ- ments with past and future boundary conditions,” ing quantum state diffusion by heterodyne detection of Quantum Studies: Mathematics and Foundations 6, 241 fluorescence,” Phys. Rev. X 6, 011002 (2016). (2019). [68] P. Campagne-Ibarcq, S. Jezouin, N. Cottet, P. Six, [50] Philippe Lewalle, John Steinmetz, and Andrew N. Jor- L. Bretheau, F. Mallet, A. Sarlette, P. Rouchon, and dan, “Chaos in continuously monitored quantum sys- B. Huard, “Using spontaneous emission of a qubit as tems: An optimal-path approach,” Phys. Rev. A 98, a resource for feedback control,” Phys. Rev. Lett. 117, 012141 (2018). 060502 (2016). [51] Sreenath K. Manikandan, Cyril Elouard, and An- [69] M. Naghiloo, N. Foroozani, D. Tan, A. Jadbabaie, drew N. Jordan, “Fluctuation theorems for continu- and K. W. Murch, “Mapping quantum state dynam- 25

ics in spontaneous emission,” Nature Communications [89] Pierre Rouchon and Jason F. Ralph, “Efficient quantum 7, 11527 (2016). filtering for quantum feedback control,” Phys. Rev. A [70] M. Naghiloo, D. Tan, P. M. Harrington, P. Lewalle, 91, 012118 (2015). A. N. Jordan, and K. W. Murch, “Quantum caustics in [90] Fernando Quijandr´ıa, Ingrid Strandberg, and G¨oran resonance-fluorescence trajectories,” Phys. Rev. A 96, Johansson, “Steady-State Generation of Wigner- 053807 (2017). Negative States in One-Dimensional Resonance Fluo- [71] D. Tan, N. Foroozani, M. Naghiloo, A. H. Kiilerich, rescence,” Phys. Rev. Lett. 121, 263603 (2018). K. Mølmer, and K. W. Murch, “Homodyne monitoring [91] Tom´aˇsTyc and Barry C. Sanders, “Operational formu- of postselected decay,” Phys. Rev. A 96, 022104 (2017). lation of homodyne detection,” Journal of Physics A: [72] M. Naghiloo, D. Tan, P. M. Harrington, J. J. Alonso, Mathematical and General 37, 7341 (2004). E. Lutz, A. Romito, and K. W. Murch, “Heat and work [92] Jacek Dziarmaga, Diego A. R. Dalvit, and Wojciech H. along individual trajectories of a quantum bit,” Phys. Zurek, “Conditional quantum dynamics with several ob- Rev. Lett. 124, 110604 (2020). servers,” Phys. Rev. A 69, 022109 (2004). [73] Q. Ficheux, S. Jezouin, Z. Leghtas, and B. Huard, “Dy- [93] P. M. Harrington, D. Tan, M. Naghiloo, and K. W. namics of a qubit while simultaneously monitoring its Murch, “Characterizing a statistical arrow of time in relaxation and dephasing,” Nat. Comm. 9, 1926 (2018). quantum measurement dynamics,” Phys. Rev. Lett. [74] Quentin Ficheux, “Quantum trajectories with incom- 123, 020502 (2019). patible decoherence channels,” Dissertation, Ecole´ nor- [94] Philippe Lewalle, Cyril Elouard, Sreenath K. Manikan- male sup´erieure—Paris (2018). dan, Xiao-Feng Qian, Joseph H. Eberly, and Andrew N. [75] Mahdi Naghiloo, “Introduction to experimental quan- Jordan, “Entanglement of a pair of quantum emitters tum measurement with superconducting qubits,” arXiv under continuous fluorescence measurements,” arXiv: 1904.09291 (2019). 1910.01206 (2019). [76] Areeya Chantasri, “Stochastic path integral formalism [95] Yui Kuramochi, Yu Watanabe, and Masahito Ueda, for continuous quantum measurement,” PhD Disserta- “Simultaneous continuous measurement of photon- tion, University of Rochester (2016). counting and homodyne detection on a free photon field: [77] N. Bergeal, F. Schackert, M. Metcalfe, R. Vijay, dynamics of state reduction and the mutual influence of V. E. Manucharyan, L. Frunzio, D. E. Prober, R. J. measurement backaction,” Journal of Physics A: Math- Schoelkopf, S. M. Girvin, and M. H. Devoret, “Phase- ematical and Theoretical 46, 425303 (2013). preserving amplification near the quantum limit with a [96] Ivonne Guevara and Howard Wiseman, “Quantum state Josephson ring modulator,” Nature 465, 6 (2010). smoothing,” Phys. Rev. Lett. 115, 180407 (2015). [78] Emmanuel Flurin, “The Josephson Mixer: A Swiss [97] Areeya Chantasri, Ivonne Guevara, and Howard M army knife for microwave quantum optics,” PhD Dis- Wiseman, “Quantum state smoothing: why the types of sertation, Ecole Normale Sup´erieure,Paris (2014). observed and unobserved measurements matter,” New [79] Ananda Roy and Michel Devoret, “Introduction to para- Journal of Physics 21, 083039 (2019). metric amplification of quantum signals with Joseph- [98] Ivonne Guevara and Howard M. Wiseman, “Completely son circuits,” Comptes Rendus Physique 17, 740 – 755 positive quantum trajectories with applications to quan- (2016). tum state smoothing,” arXiv:1909.12455. [80] Heinz-Peter Breuer, Bernd Kappler, and Francesco [99] S. W. Hawking, “Arrow of time in cosmology,” Phys. Petruccione, “Stochastic wave-function approach to the Rev. D 32, 2489–2495 (1985). calculation of multitime correlation functions of open [100] Lorenzo Maccone, “Quantum solution to the arrow-of- quantum systems,” Phys. Rev. A 56, 2334 (1997). time dilemma,” Phys. Rev. Lett. 103, 080401 (2009). [81] Claude Cohen-Tannoudji, Jacques Dupont-Roc, and [101] Joel L. Lebowitz, “Boltzmann’s entropy and time’s ar- Gilbert Grynberg, Atom—Photon Interactions (1998). row,” Physics Today 46, 32 (1993). [82] H.-P. Breuer and F. Petruccione, The Theory of Open [102] Andrew N. Jordan and Alexander N. Korotkov, “Un- Quantum Systems (Oxford University Press, 2002). collapsing the wavefunction by undoing quantum mea- [83] K. W. Murch, U. Vool, D. Zhou, S. J. Weber, S. M. surements,” Contemporary Physics 51, 125–147 (2010). Girvin, and I. Siddiqi, “Cavity-Assisted Quantum Bath [103] Alexander N. Korotkov and Andrew N. Jordan, “Un- Engineering,” Phys. Rev. Lett. 109, 183602 (2012). doing a weak quantum measurement of a solid-state [84] Howard J. Carmichael, Statistical Methods in Quantum qubit,” Phys. Rev. Lett. 97, 166805 (2006). Optics I: Master Equations and Fokker–Planck Equa- [104] Vladimir Y. Chernyak, Michael Chertkov, and Christo- tions (Springer–Verlag Berlin Heidelberg, 1999). pher Jarzynski, “Path-integral analysis of fluctuation [85] R. P. Feynman, “The Brownian Movement,” in The theorems for general Langevin processes,” Journal of Feynman Lectures on Physics, Volume I (CalTech) Statistical Mechanics: Theory and Experiment 2006, Chap. 41. P08001 (2006). [86] C. W. Gardiner, Handbook of Stochastic Methods for [105] D. Tan, S. J. Weber, I. Siddiqi, K. Mølmer, and K. W. Physics, Chemistry and the Natural Sciences (Springer, Murch, “Prediction and retrodiction for a continuously 2004). monitored superconducting qubit,” Phys. Rev. Lett. [87] Peter E. Kloeden and Eckhard Platen, Numerical So- 114, 090403 (2015). lution of Stochastic Differential Equations (Springer, [106] Nathana¨elCottet, S´ebastienJezouin, Landry Bretheau, 1992). Philippe Campagne-Ibarcq, Quentin Ficheux, Janet An- [88] William H. Press, Brian P. Flannery, Saul A. Teukol- ders, Alexia Auff`eves, R´emiAzouit, Pierre Rouchon, sky, and William T. Vetterling, Numerical Recipes in and Benjamin Huard, “Observing a quantum Maxwell C (Cambridge University Press, 1988). demon at work,” PNAS 114, 7561–7564 (2017). [107] Cyril Elouard, David A. Herrera-Mart´ı,Maxime Clusel, 26

and Alexia Auff`eves, “The role of quantum measure- tum error correction through local operations,” Phys. ment in stochastic thermodynamics,” NPJ Quantum In- Rev. A 82, 032327 (2010). formation 3, 9 (2017). [124] Andr´e R. R. Carvalho and Marcelo Fran¸ca Santos, [108] Cyril Elouard, David Herrera-Mart´ı,Benjamin Huard, “Distant entanglement protected through artificially in- and Alexia Auff`eves, “Extracting Work from Quantum creased local temperature,” New Journal of Physics 13, Measurement in Maxwell’s Demon Engines,” Phys. Rev. 013010 (2011). Lett. 118, 260603 (2017). [125] Howard M. Wiseman, “Death by experiment for local [109] Cyril Elouard and Andrew N. Jordan, “Efficient quan- realism,” Nature 526, 649 (2015). tum measurement engines,” Phys. Rev. Lett. 120, 260601 (2018). [110] Y. Masuyama, K. Funo, Y. Murashita, A. Noguchi, S. Kono, Y. Tabuchi, R. Yamazaki, M. Ueda, and Y. Nakamura, “Information-to-work conversion by Maxwells demon in a superconducting circuit quantum electrodynamical system,” Nature Communications 9, 1291 (2018). [111] Juliette Monsel, Cyril Elouard, and Alexia Auff`eves, “An autonomous quantum machine to measure the ther- modynamic arrow of time,” npj Quantum Information 4, 59 (2018). [112] Juliette Monsel, Marco Fellous-Asiani, Benjamin Huard, and Alexia Auff`eves, “A coherent quan- tum engine based on bath and battery engineering,” arXiv:1907.00812 (2019). [113] M. H. Mohammady, A. Auff`eves, and J. Anders, “Energetic footprints of irreversibility in the quantum regime,” arXiv:1907.06559 (2019). [114] Jin Wang and H. M. Wiseman, “Feedback-stabilization of an arbitrary pure state of a two-level atom,” Phys. Rev. A 64, 063810 (2001). [115] Kater W. Murch, Rajamani Vijay, and Irfan Siddiqi, “Weak measurement and feedback in superconducting quantum circuits,” arXiv:1507.04617v2 (2015). [116] Juan Atalaya, Shay Hacohen-Gourgy, Leigh S. Martin, Irfan Siddiqi, and Alexander N. Korotkov, “Multitime correlators in continuous measurement of qubit observ- ables,” Phys. Rev. A 97, 020104 (2018). [117] Eduardo Mascarenhas, Daniel Cavalcanti, Vlatko Ve- dral, and Marcelo Fran¸caSantos, “Physically realiz- able entanglement by local continuous measurements,” Phys. Rev. A 83, 022311 (2011). [118] M. F. Santos, M. Terra Cunha, R. Chaves, and A. R. R. Carvalho, “Quantum computing with incoherent re- sources and quantum jumps,” Phys. Rev. Lett. 108, 170501 (2012). [119] Philippe Lewalle, Cyril Elouard, Sreenath K. Manikan- dan, Xiao-Feng Qian, Joseph H. Eberly, and Andrew N. Jordan, “Diffusive entanglement generation by contin- uous homodyne monitoring of spontaneous emission,” arXiv: 1910.01204 (2019). [120] Ting Yu and J. H. Eberly, “Finite-time disentanglement via spontaneous emission,” Phys. Rev. Lett. 93, 140404 (2004). [121] Florian Mintert, Andr´eR. R. Carvalho, Marek Ku´s, and Andreas Buchleitner, “Measures and dynamics of entangled states,” Physics Reports 415, 207 – 259 (2005). [122] Carlos Viviescas, Ivonne Guevara, Andr´eR. R. Car- valho, Marc Busse, and Andreas Buchleitner, “En- tanglement dynamics in open two-qubit systems via diffusive quantum trajectories,” Phys. Rev. Lett. 105, 210502 (2010). [123] Eduardo Mascarenhas, Breno Marques, Marcelo Terra Cunha, and Marcelo Fran¸caSantos, “Continuous quan-